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^LIBRARY OF CONGRESS.^ 

f [SMITHSONIAN DEPOSIT.] ' 



? UNITED STATES OF AMERICA.! 



THE 



PRACTICAL SURVEYOR'S GUIDE, 



THE NECESSARY INFORMATION TO MAKE ANY PERSON 
OF COMMON CAPACITY, 



A FDaSHED LAND SIJRYEYOE, 



WITHOUT THE AID OF A TEACHER. 



BY 

ANDREW DUNCAN, 

LAND SURVEYOR AND CIVIL ENGINEEB. 



iDl 



PHILADELPHIA: 

HENRY CAREY BAIRD, 

(successor to e. l. caret.) 

LONDON: 

LOW, SON & CO. 

1854. 



Entered, according to act of Congress, in the year 1854, by 

HBNRY^CARET BAIRD, 

In the Clerk's OfiSce of the District Court of the United States, for 

the Eastern District of Pennsylvania. 



STEBEOTYPED BY SLOTE & MOONEY, 

Philadelphia. 



^^p 



s-s-siso 



ADYEETISEMENT. 



The intention of the Author of the following 
synopsis is to furnish a cheap, small book contain- 
ing the best practical information hitherto pub- 
lished and scattered through many eminent authors, 
nearly all of which he has carefully studied, and 
having had more than thirty years' experience as a 
Surveyor, &c., has often had occasion to carry 
large works in order to have at hand the things 
necessary in the immediate field practice of sur- 
veying, levelling, profiling, calculating excava- 
tions, embankments, &c. To render the book as 
cheap as possible, the tables shall be only those 
that are really wanted, but- by which all things in 
common practice can be readily done, viz : Lati- 
tude and departure, for four pole chains. A table 
of natural sines, cosines, and tangents, which are 
most useful for finding the radius of curves, degree 
of curvature, &c., on railroad and other curves, 
their application in finding grades, cuttings, fil- 
lings, &c., will, it is hoped, prove useful. 

(iii) 



iv Advertisement. 

Geometrical demonstrations are avoided, except 
in a few cases, the design of the writer being to 
furnish a short treatise, to direct at once to what 
is required. 

The work will be comprised in four divisons : 

1st. The arithmetical calculation of plane figures. 

2nd. The calculation of surveys taken with the 
compass and chain by latitude and departure, with 
various methods of proof. 

3d. The method of plotting, enlarging and di- 
minishing maps, with remarks on copying and 
embellishing. 

4th. Levelling, profiling and calculating cuttings 
and embankments, the use and application of the 
tables, together with many other useful things 
applicable in practice. 



PREFACE. 



The following compilation is made in consequence of the 
undersigned not having met with any work on Surveying 
sufficiently concise, and instructive in the several details, 
necessary to qualify properly the practical surveyor. Many 
of the works already published contain subjects not neces- 
sary in such treatises ; such as Geometry, Plane Trigo- 
nometry, &c., which subjects, it is taken for granted, all 
who intend to become proficients have studied prior to 
reading Surveying. They are also found not to contain 
instruction that in recent improvements the surveyor re- 
quires to know. Many of these things the compiler of this 
short treatise, will endeavour to supply ; also, many other 
necessary things, which, in his long experience, he has 
found indispensable to the correct practitioner. He has 
collected the most necessary insti-uction in leveling and 
profiling, with a new and speedy plan of setting grades 
on rail and plank roads. The method of inflecting curves, 
not hitherto sufficiently explained. The description and 
design of a new instrument whereby distances are found 
at once without any calculation, A new method of sur- 
veying any tract of land by measuring one line through 
it, with a geometrical demonstration of the same. A geo- 
metrical method of correcting surveys taken with the Com- 
1* (v) 



vi Preface. 

pass, to fit them for calculation, with a table of corrections 
for certain distances, but applicable to all. A short method 
of finding the angles from the courses, and vice versa. 
The method of surveying with the Compass through any 
mine or iron works, and to correct the deflections of the 
needle by attraction. Description of an instrument by the 
help of which any gentleman may measure a map by inspec- 
tion, without calculation. A new and short method of calcu- 
lation, wherein fewer figures are used than in the common 
method ; also, the Pennsylvania method. Tables of diff'er- 
ence ■ of Latitude and Departure, made expressly for two 
pole chains, but which can also be used with four poles. 
The method of correcting the diurnal variation of the 
needle, most useful in tracing the boundaries of surveys, 
a complete description of which is given with the reason 
for using 57-3° and how it is found. Various methods of 
plotting and embellishing maps. The most correct method 
of laying ofi" lots with a pole, plummets, &c. Description 
of a new Compass which the compiler has contrived for 
that purpose, and which is made by Reid & Sons, Smith- 
field street, Pittsburgh. 

The compiler does not deny that he has borrowed from 
many authors those things he has found best adapted to 
the completion of a work adequate to make a finished 
American Surveyor, of which an unprejudiced and en- 
lightened public are the best judges. 

ANDREW DUNCAN, 

Land Surveyor and Civil Engineer, 

Office, Odeon Building, Fourth St., Pittsburgh, 



PRACTICAL SUEYEYOE'S GUIDE. 



Problem First. 
To reduce two pole chains and links to four pole 
ones. 

If the number of chains be even, the half of 
them will be four pole ones, to which annex the 
given links. Thus : 

1. In 16 chains, 37 links of two pole chains 
how many four pole ones : 

2)16. 37 



Ans. 8.37 

But if the number of chains be odd, take half 
of them and add 50 to the links. Thus : 
2)131.40 



Ans. 65.90 

Problem Second. 
To reduce four pole chains and links to two pole 
ones. Double the chains and annex the links if 

(9) 



10 The Surveyor's Guide. 

they be less than 50, but if they exceed 50, add one 
to double the chains and take 50 from the links. 

C, L. 

Thus : 16. 25 of four poles, how many two pole 
chains. 

16.25 . 

2 



Ans. ^32.25 

C. L. 

2d. In 19.87 four pole chains how many two 

pole ones. 

: 19.87 
2.50 



Ans. 39.37 

To reduce two pole chains and links to perches 
and decimal of a perch, multiply the chains by 

C. L. 

two and the links by four, thus: In 16.37 how 

many perches. 

16.37 
2. 4 



Ans. 33.48 

Article First — of Areas. 
A square is a plane figure having four equal 
sides and four right angles. To find the contentj 



The Surveyor's Gtuide. 11 

multiply the side into itself aB;d the product is the 
content. 

Example. 
Required the area of the square A B C D, one 
of whose sides is 25 chains 95 links. 



Fig. 1. 



25.95 
25.95 

12975 
23355 
12975 
5190 


A 


A. K. P. 

67. 1. 14.44 


A. 67.34025 
4 


C 


25.95 


R. 1.36100 
40 




P. 14,44000 







A parallelogram is a four sided figure whose 
opposite sides and angles are equal. To find the 
area multiply one of the sides by the p.erpendicular 
demitted from one of its opposite, angles. 
Example. 

Required the area of the parallelogram A B 



12 The Surveyor's Guide. 

C D, the length of which is 15 chains, and height 

C. L. 

12 64. 



Fig. 2 



12.64 


A. R. p. 

18.3. 33.6 


D 


15 00 ( 

12.64 
15 




6320 
1264 


Acres, 


18,960 


Roods, 


3.840 
40 



Perches, 38,600 

The content of an oblong piece of ground and 
one side are frequently given to find the other. 
Divide the area in perches by the given side, gives 
the side required which is easily reduced into 
chains and links. 

If a lot contains 507 perches and is 14.25 long, 
what is its width. 



The Surveyor's Guide. 13 

29)507 '17.4827 
29 



.25+12.06=8 37.06 



217.0000 
203 

140 
116 

240 
232 

80 
58 

220 

203 
To draw maps of these figures is too obvious to 
require any explanation. 

5th. When the sides of the above figures are 
given in feet and inches, reduce the inches to deci- 
mal of a foot. Then multiply the length by the 
breadth and divide the product by 43560, the 
number of feet in an acre, the quotient will be the 
acres and decimal of an acre, which may be redu- 
ced to roods and perches by multiplying by 4 for 
the roods and 40 for the perches, pointing off the 
proper number of decimal places each time, thus : 

A lot of land is 600 feet 4 inches long and 240 
2 



14 The Surveyor's Guide. 

feet 3 inches wide, how many acres does it contain. 
600.333.X240.25=144230.00325 

This divided by 43560. gives 3.31106 or 3 1 0^9.76 

4 



1.24424 
40 



9.76960 Ans. 

6th. A trapezium is a four sided figure the oppo- 
site sides of which are neither equal nor parallel. 
To find the content, measure a diagonal and two 
perpendiculars to the opposite corners, multiply 
the diagonal by half the sum of the perpendicu- 
lars, and the product will be the area. 
Example. 

Let A B C D be any trapezium, having A C 80 
perches, and the perpendiculars as in the figure. 




The Surveyor's Guide. 



15 



25 
20 

2)45 

22.5 
80 

160)1800,0(li.l.00 Ans. 
160 

200 
160 

40)40(1 
40 



7th. A triangle is a figure having three sides 
and three angles, any side may be called the base, 
having the base and perpendicular given. Multi- 
ply the base by half the perpendicular, or the base 
by the whole of the perpendicular, and take half 
the sum. 

Example. 

LetABCbeany 
triangle whose base 
is 100 two pole ch's 
and 15 links, and 
perpendicular 40 




16 The Surveyor's Guide. 

chains and 20 links, required the content in 

acres. 

100 15 
2 4 



40 20 200.6 perches and decimal. 

2 4 40.4 half the perpendicular. 



2)80 8 8024 

8024 

40 4 A. R. P. 

160)8104.24(50 2 24.24 Ans. 
800 

40)104 
80 

24 

8th. Having the three sides given to find the area 
rule, add the three sides together and take half 
the sum, from which subtract each side severally, 
multiply the half sum and three remainders con- 
tinually into each other, and the square root of 
the product will be the area. 

The most satisfactory proof of the above rule 
is the following : 

Let A B C be any triangle, B C its base, A B the 
greatest side, and A C the least, and let P be half 




The Surveyor's Guide. 17 

the perimeter. In A B take A D=A C, join D C 
and draw A E perpendicular to D C and E G pa- 
allel to B C cutting A B 
in G, with tlie centre G 
and radius G E describe 
a circle cutting A B in L 
and A B and E G pro- 
duced in K and H. Join 
H B and produce A E, Kg. 5. 

H B till they meet in M. Since A D=A C and 
the angles at E are right, the squares of A E, 
E D are equal 2 A E,+2 E and .-. E D=E C or 
D 0=2 D E. Hence by similar triangles D G E, 
D B 0, B 0=2 G E=E H, and B is also paral- 
lel to E H .-. H B M is parallel to E D and 
(Euc. 1st 29th) the angle B M E=D E A, viz : a 
right angle, and H E being a diameter, M is a 
point in the circle. But from similar triangles 
D G E, D B 0, D B=2 D G, to each of these 
equals add A D+A 0=2 A D, and B A+A 0=2 
A G, to each of which equals add B 0=2 G E=2 
G K .-. A B+A 0+B 0=2 A K or A K is =P, 
half the perimeter. Now the area of the triangle 
A B 0=area A D 0+area B D 0=A EXD E+ 



18 



The Sueveyor's Guide. 



M EXD E (B M being parallel to D C)=A MXD 
E. But by similar triangles A D E, A M B : A 
E : E D :: A M to M B, and by equi-multiples 
the first multiplied by third : the second multiplied 
by the third :: the second X by the third : the 
second X by the fourth. Hence, A EXA M : E 
DXA M :: E DXA M : E DXM B, i. e. the area 
of the triangle is a mean proportional between 
A EXA M, and E DXM B. Now E DXM B = 
P— A BXP— A C, and A EXA M=A LXA K= 
PXP — B 0. Hence the area of the triangle is : 
^fPXP— A BXP— A GXP— B 0, which is the rule. 

Example. 
9th. Suppose the 
sides to be measur- 
ed by a four pole 
chain and be 

AB 10.64) 
A C 12. 28 y 
BO 9. 00 J 



Sum 31.92 



I sum 15. 96 

5. 32 first remainder. 
3. 68 second do. 

6. 96 third do. 




Fig. 6 



The Surveyor's Guide. 19 



15.96X5.32X3.68X6.96=2174.71013216(46.6337 
16 





A. R. 

4 2 


616 






926)5871 
5556 


And since 
10 square 
4 pole ch's 
make one 
acre, this 
becomes 




9323)31501 
27969 




93263)353232 

279789 




932667)7344316 4.66337 
6528669 4 




p. 
26. 


2.65348 
40 


The content is 


26,13920 



If the sides are in perches and decimal, divide 
the square root of the products of the half sum 
and three remainders by 160, and the quotient 
will be the acres, and the remainder divided by 40 
will be the roods. 

The same may be more readily done by loga- 
rithms, for as the addition of logarithms serves for 



20 



The Surveyor's Guide. 



the multiplication of their corresponding numbers, 
and that the number answering to the half of a lo- 
garithm will give the square root of the number of 
that logarithm, it follows that half the sum of the 
logarithms of half the sum of the sides, and the 
three remainders will give the area, thus : 
Half sum, 15.96 log. 1.20303 



First remainder, 5.32 
Second " 3.68 
Third « 6.96 


" 0.72591 
" 0.56585 
" 0.84261 

2)3.33740 


Square four poles 46.63 
Or, 4.663 
4 


1.66870 


2.652 
40 




26.080 4 2 26 as before. 



10th. When the three sides are given and the 
angles are required, call either side on which the 
perpendicular will fall from the o^pposite angle the 
base, then as the base is to the sum of the other 
two sides so is the difference of those sides to the 
difference of the segments made by the perpen- 
dicular, then half that difference added to half the 



The Surveyor's Guide. 



21 



Bum gives the greater, and substracted the less, by 

•which means it is divided into two right angled 

triangles, the hypothenuse and one leg of each 

being given, the angles are easily found by plane 

trigonometry. 

Example. 
Let A B C be ^ \ Kg. 7 

any triangle hav- 
ing the sides giv- 
en as follows,viz : 
A B 88, B C 54 
and A C 108 to find the angles. 

A B=88 Then as 108 : 142 :: 34 
B C=54 34 




142 sum 
34 difference. 



568 
426 



108)4828.000(44.703 diff. of the 
432 segments at the base. 

22.351 half diff. 

508 
432 

760 
756 



400 
324 



22 The Surveyor's Guide. 

Then half the base 54+22.351=76.351, the 
greater segment A B, and 54—22.351=31.649 the 
less segment. 

The triangle is now divided into two right angled 

triangles, the hypothenuse and base in each being 

given to find the angles, as follows : 

AsAB 88 1.9444827 

: Rad. 90° 10.0000000 

:: A D 76.351 1.8828147 



11.8828147 
1.9444827 



: Sine A B D 60°.ll' 9.9383320 

And 90— 60°.11'=29°.49'=-Angle BAD. In 
the same way C B D is found to be 35°. 53' its com- 
plement 54°.07'=/' B CD. 

Now A B D=60°.ll' 
C B D=35 .53 



Angle A B C= 


=96 


.04 


t 


A= 


29 


.49 


L 


C- 


54 


.07 



180 .00 Proof as the three angles 
of every plane triangle are equal to 180° per 32d 
of the 1st of Euclid. 

lltli. Many things occur to the practical sur- 



The Surveyor's Guide. 23 

veyor in the triangle, some of which I shall take 
notice of in this place. It often happens in prac- 
tice that the two sides and their included angle are 
given to find the other angles and side. 

Rule. — As the sum of the sides is to their differ- 
ence so is the tangent of half the sum of the oppo- 
site angles to the tangent of half the difference ; 
this half difference added to half the sum of the 
angles at the base gives the greater, and sub- 
tracted the less. Then aa sine of either of the 
base angles is to its opposite side, so is sine of the 
contained angle to the required side. 
Example. 



Let AC 


=80, B G 


=110,and 


/^ ACB 


102°.30'to 


find A B 




c 
and the angles A and B. 

Side B C 110 From 180 
Side AC 80 take /: C= 102.30 



Sum 190 2)77.30 sum of base 

angles. 

Diff. of sides 30 | sum -38.45 



24 



The Surveyor's Guide. 



Then as 190 log. 2.2787536 88°.45' 
: 30 " 1.4771213 7 .13 

:: Tag't 38°.45' 9.9044910 

45 .58 U A 

11.3816123 — 

2.2787536 31 .32 B 



::Tag'tofidiff.7°.13' 9.1028587 

Then as sine B 31°.32' 

: AC 80 

:: sine C 102.30 
Or its supplem't 77.30 



9.7184971 
1.9030900 

9.9895815 

11.8926715 

9.7184971 



To A B 149.34 2.1741744 

12th. Again, it often happens that the area 
must be found from the foregoing data, in that 
case multiply the two sides together, and that pro- 
duct by the natural sine of \ the contained angle, 
gives the area. 

Example. 

Let ABC 

be a triangle 

having the 

side A C 13 

chains, A B 4. 

7c. 501. and /- B A C 42° to find the area. 




The Surveyor's Guide. 25 

7.5 
13 

225 
75 

97.5X.334565 half the nat'l sine of 42°= 

A. R. P. 

32.62 square four pole chains = 32 19.2 Ans. 

Demonstration. 
Let fall the perpendicular B. D. 
A B : B D :: rad : sine A 
.-. B D=ABXsine^A 



Rad But rad.=l. 

.*. B D=A B=sine. Multiply each side by 
ACandBD. AC=AB. Sine A.XA C 
But A C. B D— the area. Hence, 



A B. A C. Sine A=area, which is the rule. 



13th. Let B A C be a triangular farm, and P a 
well of water. It is required to draw a line or 
fence from the well that will divide the farm 
equally between two partners; 



26 



The Surveyor's Guide. 

Fig. 10. 




E D 

Find D the middle of the base, B C, and from 
P take a course of P D. Again set your instrument 
at A, and take the same course A E ; cause a pole 
to be set at E, a line or fence from E to P will 
bisect the farm, which is easily demonstrated from 
the figure. See Bland. 

14th. Again, suppose the well P, to be situated 
within the farm, and it be required to divide it 
equally between three occupants, so that each may 
have the use of the well. a 

In fig. 11 di- 
vide the base B 
C, into three 
equal parts in 
D and E. Set 
your instrument 
at P, and take ^ 




Fig. 11. 



The Suevetor's Guide. 



27 



the courses P D and P E. Remove your instru- 
ment to A, and take A F the same course as P D, 
and A G the same as P E. Cause stakes to be driven 
at F and E in a straight line between B and C. 
Fences from F, G, and A, to P, trisect the farm, 
■which is plain from the figure. 

15th. To find the area of a Trapezoid Rule, 
multiply half the sum of the parallel sides by the 
perpendicular distance between them, and the pro- 
duct is the area. 

Let figure 12 be a 
Trapezoid ; if A D be q ..-,- 
bisected in E, and E 
F drawn parallel to 

A B or C D, it also j Kg. 12. 

bisects B C in F. — 
Through F draw ^G 
H parallel to A D. 




It is evident the triangles, B F G, and F C H, 
are similar and equal. (26th Euclid, 1st.) .-. E F, 
half the sum of the sides, multiplied by the per- 
pendicular distance between them, A B, gives 
the area. 



28 



The Surveyor's Guide. 



Being surveying on the side of a bog, and want- 
ing four acres to make up a division, and seeing A 




A 50 , C 

B would pass through a pond, I found A C fifty 
chains, and L C 56*^; how far must I measure 
from C towards B, so that the triangle ABC, 
may contain four acres. 

Since A C X C B X i the natl. sine of 56°=4 
acres, it follows that 4 acres divided by the pro- 
duct of one half the natl. sine of 56° into A C, 
gives B C the required side. Thus : 

50X4=200 perches X, 4145188=82.9 ; and 640 
perches in 4 acres, divided by 82.9=7.72 per the 
length of B C, and in like manner any other simi- 
lar case can be done. 

17th. Sometimes it is found necessary to ob- 
tain the area of a trapezium from having the 
diagonals and the angle of intersection given. 



The Surveyor's Guide. 29 

Rule — Half the product of the diagonals multi- 
plied by the natural sine of the angle of intersec- 
tion, will be the area. 

Example. 
If the two diagonals of a trapezium be 40.15, 
and 60.13 chains the /- of intersection 75° 45', 
what is the area. J of 40.15X60.13=1207.1097= 
half the product of the diagonals, and 1207.1097X 
96923=(natural sine of 75°45')=1169.966934531== 

A. R. P. 

the area, in square four pole chains, or 116. 3. 39. 47. 
Answer. 

18th. To find the area of a trapezium, when 
each side and the angle of intersection of the 
diagonals are given. Rule — Square each side of 
the trapezium ; add together the squares of each 
pair of opposite sides ; subtract the less from the 
greater; multiply the diflference by the tangents 
of the angle of intersection. One fourth of the 
product will be the area. 

Example. 

What is the area of a trapezium, the sides of 
which are 10, 13, 7.16, 8.32, and 10.05 chains 
respectively, and the V of intersection of the 
diagonals 52° 15'. 



30 The Surveyor's Guide. 

(10.13)=102.6169 
( 8.32)= 69.2224 

171.8393=Sum of sqs. of opposite sides. 

(15.05)=226.5025 
( 7.16)= 51.2656 



277.7681=Suni of sqs. of other sides. 

105.9288 Difference, 
Multiplied by .32288=1 the natural tangent, 

34.20290944 or 



A 3. 1 .27,23 perches. 

For a demonstration of the foregoing, see Cfib- 
son's Surveying, hj Trotter. 

19th. To find the area of a trapezium, when the 
four sides are severally given, and also the sum 
of any two opposite angles. B,ule — From half the 
sum of the four given sides, subtract each seve- 
rally ; multiply the four remainders continually 
together ; from the result subtract one half the 
continual product of the four sides, multiplied by 
unity, increased by the natural cosine of the sum 
of the given angles. The square root of the 
result will be the area. 



The Surveyor's Guide. 31 

Remark. 

In the application of this theorem, it must be 
carefully remembered that the cosine of an angle ' 
is positive when that angle is in either the first or 
fourth quadrants, and negative when it is in the 
second or third quadrants. For a demonstration 
of this beautiful theorem, see also, Grihson, hy 
Trotter. 

N. B. When the sum of the opposite angles is 
180°, that is, when the trapezium can be inscribed 
in a circle, the above rule is simply : frop half the 
sum of the given sides, subtract each side seve- 
rally ; multiply the four remainders continually 
together, and extract the square root, gives the 
area. 

Example. 

" One morning in May I went to survey, 

As soon as bright Sol I espied ; 

I measured round a four cornered ground. 

In the margin see the length of each side ; 

The angle at B, together with D, 

An hundred and fifty degrees ; 

The meadow's content is all that I want, 

Assist me kind youths, if you please." 



32 



The Surveyor's Guide. 



A B 15.60 
B C 13.20 
C D 10.00 
D A 26.00 ch'ns. 



2)64.80 sum. 



32.4=J sum. 




rig. 14. 



16.80=lst remr. S=half the sum. 

19.20=2d do. of the sides. 

22.40=3d do. 
6.40=4th do. 

Whence (s-A B)X(s-B C)X(s-C D)X(s-D A)= 
32.4X16.8X19.2X22.4X6.4-46242.2016=46242. 
2016 
And A B. B C. C D. D A.X(l+cos. 150°) 



That is 15.60X13.20X10.00X26.00 



-X0.1339746=3586.4464 



Difference=42655.7552 
The square root of 42655.7552 is 206, 5327= 

A. R. P. 

area in square four pole chains, or 20. 2. 24,55232. 

N. B. This problem is taken from Deighan's 

Arithmetic, vol. second, page 148, and the answer 

A. K. p. 

there given is 21. 2. 00,64, which is obtained by 
taking the trapezium to be inscribed in a circle, 
which is not the case. 



The Surveyor's Guide. 33 

When the opposite angles of a quadralateral are 
equal to two right angles, a circle can be described 
about it. The rule to find the area, then, is : mul- 
tiply the half sum, and four remainders continually 
together, and extract the square root, for in that 
case l+cos.(A+B)=0. 

21st. To find the area of a circle having the 
diameter given. Rule — Square the diameter, and 
multiply by .7854, and you have the area. 

22d. To find the area of an ellipsis. Rule — 
Multiply the transverse and conjugate diameters 
together, and that product by .7854, and you have 
the area. 

28d. To find the area of a parabola. Rule — 
Multiply the height by the breadth, and take two- 
thirds of the product ; you have the area. 

24th. To find the area of a segment of a para- 
bola. Rule — Multiply the base of the segment by 
the altitude thereof, and two-thirds of the product 
gives the area. 

25th. To find the area of a field or lot, which is 
found to be the frustum or zone of a parabola, 
included by two parallel right lines, and the inter- 
cepted curves of the parabola. Rule — Add the two 



34 The Sukveyor's Guide, 

parallel ends, divide the square of either of these 
ends by this sum, add the quotient to the other 
end, multiply this sum by the altitude of the frus- 
tum or distance of the ends, take two-thirds of the 
product, and it gives the area. 



TRIGONOMETEICAL SURVEYING. 



26th. It was not my intention to say any thing 
concerning this branch of surveying, as it is too 
extensive a subject for this small work ; but as 
some young readers may not have met with any 
thing on that subject, I will present them with an 
outline of how that grand operation is conducted. 

When an entire country, or part of a country, 
containing one or more counties is to be surveyed, 
it is done by triangulation, and the application of 
the rule given in the 12th section of this work. A 
line of some miles in length is measured and re- 
measured in order to prove its accuracy, on some 
plane or heath which is nearly level, first having 
been traced by a transit instrument, and poles 
placed in an exact straight line, to guide the meas- 
urers, as A B in the annexed figure, which is assu- 
med as the base of the operations. A number of 
hills and elevated spots are selected, on which sig- 
nals can be placed, suitably distant and visible 

(35) 



36 



The Surveyor's Guide. 



one from another. Thus, ifACDEBHG 
F, &c., be several objects, the situations of which 

D 

Fig. 15. 




are to be laid down on a map, and they are 
within the lines, ACDEBHGF, accu- 
rately calculated. It is supposed that the stations 
A and B are chosen such as that all the others can 
be seen from each of them. Then from the ex- 
tremity A, measure the angles E A B, D A B, 
CAB, &c., H A B, G A B, F A B, &c. And 
from the other extremity B, measure the angles, 
CBA,DBA,EBA, &c.,FBA, GBA,HBA, 
&c. And as the common base, A B, and the seve- 
ral angles of all the triangles are now known, the 
sides, A C, A D, A E, &c. may be determined by 
simple proportion, for as the natural sine of 



The Surveyor's Guide. 37 

A C B : A B : : sine C A B : C B and so is sine ABC 
to C A, and so through all the triangles, the three 
sides being thus found in each triangle, the area is 
easily found, as shown in section 8th of this trea- 
tise. But to insure accuracy the objects C D E, etc., 
should be all intersected from some third station, 
O in the base A B, otherwise the figure may ap- 
pear in the plotting to be right when it is not so, 
and there will be no means of knowing whether 
the angles have been correctly taken without going 
over the work again. 

27th. Here follows an example of a triangle con- 

A. R. p. 

taining a mean area of 1135.2.12.79. The] sides 
of which were traced by a transit instrument, and 
poles placed at the several points marked thus O ; 
this being done, the respective distances of the 
sides were ascertained by a mean of measures as 
follows, viz : 

B A 14643 links, or 9664.38 feet, A C 17814 
links, or 11777.24 feet, B C 16588 links, or 10948. 
08 feet. The angles were taken by a theodolite 
as they are marked in the figure. 

Now to determine the area of the triangle, A 
BC: 



38 



The Surveyor's Guide. 



50°.1S'.21', 



Fig.IO. 



/'^^ 



14643 Unks, or 9664.38 feet. 



1st. From the data, A B, and the three angles of 
the known formula 

A B^Xsine BXsine A, a r p' 

=1135.2.27.18 

2 sine C. 
2d, by B C, and the three angles, 

the area will be 1135.3.029 

3d, by C A, and the three angles, 

the area will be 1135.0.38.6 



The Surveyor's Guide. 39 

4th, by data A B, and the two 
adjacent angles, we have by the 
known formula, 

A B^Xsine BXsine A, 



2 sine (B+A) 
The area will be 1135.2.25.7 

5th, and by B C, and the two ad- 
jacent angles 1135.3.01.9 
6th, by a similar formula from A 
C, and the two adjacent angles, 
the area will be 1135.0.37.99 
7th, by data A B, and the adja- 
cent angle A, and the remote an- 
gle C, we have by the known 
formula, 

(A B)^Xsine AXsine (C+A) 

2 sine C area, 1135.2.27.8 

8th, by a similar formula from hav- 
ing A B, and the angles, B and 
C; area 1135.2.28.2 

9th, by having C B and the angles, 

C and A; area 1135.3.03.58 

10th, by having C B and the angles 

B and A: area 1135.3.04.38 



40 The Surveyor's Guide. 

11th, by a similar formula data C 
A, and the angles, C and B, gives 
the area 1135.0.39.66 

12th, by a similar formula from da- 
ta C A, and the angles A and 
B; area 1135.1.00.12 

13th, by data A BXB C, and the 
contained angle, we have 

A BXB CXsine B-1135.2.35.06 



2 

14th, by A CXA B, and the con- 
tained angle 1135.1.32.92 

15th, by A CXB C, and their con- 
tained angle C 1135.2.00.79 

16th, by data, A BXB C, and the 
angle, A, we have by a known 
formula, B AXsine A 



BC. 

=sine C, and A BXB C, sine (A+O) 

2 area 1135.2.394 

17th, by the application of similar 
formula to the data, A BXB C, 
and angle, C ; area 1135.2.30.4 



The Surveyor's Guide. 41 

18th, by A' CXC B, and angle, A, 1136.0.19.51 
19th, by A CXB C, and angle, B, 

the area will be 1135.0.19.89 

20th, by A CXB A, and angle, B, 

the area will be 1135.1.10.5 

21st, by A CXB A, and angle, C, 

the area will be 1135.3.05.16 

22d, by the usual rule from the 

three sides, s. s — a. s — b. s — c. 1135.2.14.7 

Now the various data exhibited in this triangle 
have been ascertained with the same relative de- 
gree of precision ; and the different areas deduced 
therefrom have been subjected to the same loga- 
rithmic process, till the figure has been exhausted ; 
there is no reason to suppose that any one of them 
is nearer to the truth than another ; and taking a 

A. R. p. 

mean of the results we have 1135.2.12.97 for the 
nearest approximation to the true area. 

But suppose we consider the triangle as spheri- 
cal, and the admeasurement of the sides as the 
lengths of three arcs of three great circles of the 
sphere ; and, according to Sir Isaac Newton, the 
diameter of the earth to be 41,798,177 feet, we 
will then have, as the circumference of a great 

4* 



42 The Surveyor's Guide. 

-circle of the earth is to 360°, so is the length of 
C B to the number of degrees or minutes, &c., 
contained in the arch, C B, viz : 



As 131312964.37 
And do. 
And do. 



360°:: 11757.24 : l'.56".03868=archC A. 
do.:: 9664.38 :r.35".38309= " AB 
do. :: 10948.08 : r.48".05263== « B C. 



Now let b a c, represent the sides of any spheri- 
cal triangle, and e the spherical excess, we have by 
Lhuiller's theorem. Tangent ^ E= 

Tan. a+b+c. tan. a+b — c. tan. a — b+c. tan. — a+b+c. 

i 4 ' ~4 4 

And by restoring to a b and c, their deter- 
mined values, we find 

a+b+c =0° .1' .19". 8686 

a+b— c=0° .0' .32" .1771' 



a— b+c=0° .0' .21" .8493 



And, — a+b+c=0° .0' .25" .842£ 



Whence the log. tangt. of 0° .1' .19" .8686= 6.5879531 

of .0' .32" .1771= 6.1931205 

of .0' .21" .8493= 6.0250065 

of .0' .25" .8423= 6.0979010 



2)24.9039811 



Log. of i the spherical excess= 2.4519905 
The arc corresponding to this log. will be found 



The Surveyor's Guide. 43 

to be ,00584 parts of a second, consequently the 
spherical excess is 02336 of a second, and bj a 
well known theorem, As 180° : the area of one- 
quarter the surface of the sphere : : the spherical 
excess to the area of the spherical triangle, viz : 

As 180° 31500428420,3 the area of a great cir- 
cle of the earth in statute acres : : 023360 to 

A. R. p. 

1135.2.11.3 being J perch less than the mean 
area, which is in defect, but should be in excess ; 
but this is accounted for by the hills on the land 
not being taken into account ; the difference, how- 
ever, is insignificant, and shows that the difference 
between a plane and spherical triangle of conside- 
rable dimensions is very inconsiderable. See Crib- 
son s Surveying ly Trotter. 

28th. How to measure a tract of land by measu- 
ring a base line through it, and not departing from 
that line, and yet finding all the distances round 
the land, their courses, and angles of the field, and 
the area, never before published. 

In order to do this expeditiously, the surveyor 
should be provided with an instrument having two 
telescopes, one of which is movable, and the other 
fixed, by which he can at any time take half a right 



44 The Surveyor's Guide. 

angle from the base line, and also a right angle ; 
he must also have an active assistant with a flag- 
staff, to hold at the corners as he proceeds with the 
measurement on the base line. Let ABODE 
T M G H A be any tract of land that is to be sur- 
veyed, let the base K L, be traced through it 
with a transit instrument, and poles set perpen- 
dicularly, to be visible from one to another. Set 
your instrument at L, on the base line, which in 
this survey bears N 40 E. A theodolite and com- 
pass attached is the best instrument for this 
method ; adjust your instrument, and let L be the 
point where 45° inflected from the base L K will 
cut the flag-staff; at the corner H, commence 
chaining towards K, and five chains you find 45° 
degrees deflected from the base line to the flag- 
staff at B, on the left, will bisect it, which note in 
your field-book by an oblique line to the left, ma- 
king an angle as near 45° as the eye can judge ; at 
9.30 half a right angle to the right will cut a pole 
at G, and at 12.00 came the fence ; at 13.20 half 

C. L. 

a right angle will cut C, and at 19.15 you find a 
right angle will intersect H. Now it is evident 

C. L. 

that you are 19.15 distant from H, for H 19.15. L 



The Surveyor's Guide. 



45 



is an isosceles triangle, and .*. you mark 19.15 on 
the perpendicular. The next perpendicular is at 




20, and the half right angle having been taken at 

c. 

5 on the chain line, 20 — 5=15=the distance to B. 

0. L. 

Again at 32.35 you find half a right angle bisects 



46 The Surveyor's Guide. 



the pole at D, and at 33.20 a right angle inter- 

c. L. 
sects at G, and 33.20— 9.30-=30.40=the length of 

the perpendicular which set on it. At 34.20 you 
find the next perpendicular on the left to 0, and 
the one-half right angle having been taken at 13. 
20 .-. 34.20— 13.20=21.o''o the distance to C ; pro- 
ceeding in this way you have 43.35 — 32.35=11 
chains to D, and 51.30— 35,40=15.40=the dis- 
tance to M, and 57.40—38.30=19.10 to F, and 
60— 5T.40=2.10=the perpendicular of the last A 
within the fence on the right and 62.30—60=2.30= 
the perpendicular without the fence ; also, 62.30 — 
41=21. 30=the distance to E, which A is to be 
ducted out of the area of the last trapezoid on the 
left. Thus you have found with very little trouble 
all the requisites for calculating the area of the 
land, and it may be remarked, that you might have 
commenced at the corner B and noted where the 
two perpendiculars fell at 19.15 and 20 and as you 
proceeded on your base line take back sights at 
the proper distances to intersect the poles at B and 
H, and the distances from where the perpendicu- 
lars would fall to these several points would be the 
chains and links to be placed thereon. The dis- 



The Surveyor's Guide. 47 

tances all round the land, can be accurately found, 
for in the present case n/(A a^)+(a H")= A H, 
and r Gr, and r H being given V{r G^)+(r IP)=H Gl- 
and so on all round the land, and seeing that the 
courses of A a and a H are given, the course of A 
H may be readily found, for having the distance 
and difference of latitude and departure, the course 
is given in the tables ; also, the internal angles 
can be easily found, for in the ^ A a H A a : 
Ead. : : a H : tang't a A H, and so with the A 
B b A. Hence, the angle B A H, is known, and 
it is evident the same holds good all round the 
land, the bases and perpendiculars of all the 
right angled triangles being found from the base 
line and can be marked on the sketch as the sur- 
veyor proceeds. The same may be done with a 
good compass, for having the course of the base 
line, the courses of the normals to right and left 
are known, and the course of J a right angle being 
once ascertained on the right and left of the base 
will always serve to find the points on the base 
where they are to be taken ; but this would require 
many trials and waste time, whereas, an instru- 
ment showing J a right angle will save much time. 



48 The Surveyor's Guide. 

Thus, in a plane country, the scientific reader will 
acknowledge the plan completely available, and 
the surveyor can calculate the content of the land 
on the margin of his book while his needle is set- 
tling, and be able to answer the farmer satisfac- 
torily, who thinks a surveyor should be able to tell 
the content the moment he has the last distance 
measured. 

The plotting and calculation of a survey taken 
on the above plan is so obvious as to require no 
explanation, seeing all the figures are either right 
angled triangles or trapezoids, to find the area of 
which is shown in figure 12th. 

29th. The most correct method of correcting 
the difierence of latitude and departure in surveys 
taken with the compass, to fit them for calculation, 
some authors divide the difi"erences proportionally 
among all the stations ; but as there may be some 
stations in a survey really correct, any alteration 
in them would make them incorrect, so that the 
altering of the legs of stations in surveys where 
land is of great value, is a matter of considerable 
importance. 



The Surveyor's Guide. 49 



i^ 



Pig.] 



Problem. 

To find what may be the error in the difference of latitude 
and departure of a given station arising from the inacu- 
racy of practice : 

Let the right angled triangle A B D, fig. 18th, 
represent a station "with its difference of latitude 
and departure; if the angle A be the bearing, then 
will the leg A D, be the difference of latitude, and 
the leg B D, the departure ; but if the angle at B 
be the bearing, then will the leg B D, be the differ- 
ence of latitude, and D A the departure. Let the 
small angle B A b represent the error committed 
in taking the bearing, which may amount 7|- min- 
utes, and the small part B e or E b, the error com- 
mitted in chaining, in proportion to the whole line 
AB, or A e, as 0.5 is to 5.00, (for in measuring the 
6 



50 The Surveyor's Guide. 

length of lines, there may be an error committed 
of half a link in 10 chains ; (this is found by expe- 
rience), and let e a, b d, and E c be drawn parallel 
to B D, and B n o, and res, parallel to D A. 

Case 1st. Suppose A B to be the true bearing 
and length of a station, and A b the one found by 
observation. Now it is plain that instead of the 
triangle A B D, we shall have by observation the 
triangle A b d, so that there is an error of the 
quantity n b, by which the leg B D is increased, 
and an error of the quantity B n, by which the leg 
A D is decreased, and the contrary may be sup- 
posed, if A b be the true distance and bearing and 
A B that found by observation; but when the 
angle at A is very small, D d may be supposed 
equal to (0). 

Case 2d. Suppose the true length and bearing 
of a station to be A e, and that found by measur- 
ment to be A B the bearing exact. Now it is 
plain that the leg e a is increased by the error 
r B, and that the leg A a, is increased by the error 
r e or a D, and the quantities or errors by which 
each leg is increased are in proportion to the legs 



The Suevetor's Guide. 51 

themselves, that is, B r : e a : : r e : A a, and as 
B e is to A e. 

Case Sd. Suppose A e the true bearing and 
length of a station, and A b the same, found by 
observation. This supposes a compound error both 
in chaining and bearing, and that the error in the 
bearing increases the smallest angle in respect of 
the bearing and its complement. Here we see 
that when the leg A a is increased to A D by the 
error in chaining, as in the last case, it shall, at the 
same time, be brought back to A d by the error in 
the bearing, as in case 1st. Therefore, the leg A a 
will be increased by the quantity r e — B n, or de- 
creased by the quantity B n — r e; but r e is 
greater than B n, when the angle at A is, small ; 
and B n is greater than r e, when the angle is 
near 45°; for they become equal when the angle 
is about 25°; but at the same time the leg e a will 
be increased to d b, by the error b S=b n+B r. 

Case 4th. Suppose A E the true distance and 
bearing, and A B that found by observation ; this 
supposes the error in the bearing to decrease the 
smallest angle. Now it is evident that the longer 
leg A c is increased by the error B o or D c, and 



52 



The Surveyor's Guide. 



the shorter leg decreased by the error E o. But 
B o=B n+r e (for r e=ii o) and E o=b n — B r. 
These errors are easily found in numbers by con- 
sidering the figure, and that they are always pro- 
portional to the length of the stations. 

Here follows a table of errors in links and deci- 
mals, calculated for a station of 30 two pole 
chains, and for the different angles and their com- 
plements, under which they are placed, but which 
can be changed to any other length, by altering 
them in the same proportion as are the stations. 



BAb=i°error 
in bearing. 


88 


?2 

78 


23 
67 


32 

58 


42 

48 


45 


B e 1.5 links error in chain- 
ing. 


bn= 
Bn= 


3.2 
0.0 


3.1 
0.7 


3.0 
1.4 


2.8 
1.7 


2.2 


2.3 
2.3 


Error in short Leg. ] Case 
Error in long Leg. J 1st. 


Br= 
r e= 


0.0 
1.5 


0.3 
1.5 


0.6 
1.4 


0.8 
1.3 


1.0 
1.1 


1.0 
1.0 


Error in short Leg. ] Case 
Error in long Leg. J 2d. 


bs=(bn4-Br) 
ad=(Bn^re) 


3.2 
1.5 


3.4 
0.8 


3.6 
0.0 


3.6 
0.4 


3.4 
1.1 


3.3 
1.3 


Error in short Leg. ) Case 
Error in long Leg. J 3d. 


Eo=(bn-B r) 
B o==(Bn+r e) 


3.2 
1.5 


2.8 
2.2 


2.4 

2.8 


2.0 
3.0 


1.4 
3.3 


1.3 
3.3 


Error in short Leg. 1 Case 
Error in long Leg. J 4th. 



CORROLLARY. 

Hence we may adopt the following rules for 
altering the legs of stations in the correcting of 
surveys : 



The Surveyor's Guide. 53 

Rule First. 
When the course, or angle, is either great or 
small ; or when the difference of latitude and de- 
parture are found in the beginning of the tables, 
then the shortest leg may be increased or decreased 
by any quantity not greater than 3.2 links, and 
the longest leg increased by any quantity not 
greater than 1.5 links. 

Rule Second. 

When the latitude and departure are found about 
the [middle of the tables, or when the angle is 
about 20° under or over 45°, then the shortest 
leg may be increased by any quantity not greater 
than 3.6, or rather 4 links, and the longest leg 
left unaltered, which is, when the error in the 
bearing increases the angle opposite the smallest 
side ; but when contrary, the longer leg may be 
increased by any quantity not greater than 3 
links, and the shorter leg decreased by 2 links. 
Rule Third. 

When the difference of latitude and departure 
are found in the latter part of the tables, or when 
the bearing is about 45°, then either of the legs 



54 The Surveyor's Guide. 

(they being nearly equal) may be increased or de- 
creased by any quantity not greater than 3 links, 
and the other leg by 1.4 links, but when one leg is 
increased the other must be decreased. 

These rules are on the supposition that the 
chaining is always too long, which, in practice, I 
have nearly always found to be the case ; but when 
a surveyor has reason to think otherwise, he may 
alter the rules to his opinion, not only in respect 
to this, but also relative to the quantity of the 
errors. 

A description of an instrument by which any person, though 
unskilled in surveying, may measure a map, or part of a 
map, almost at one view : 

Get a piece of good glass about 8 or 9 inches 
long, and 6 or 7 inches broad, and divide it into 
small oblong rectangles of eight-tenths of an inch 
by 5 five-tenths, as fig. 19th. By laying this in- 
strument (which I call a computor) on a map you 
can tell with very few figures, sometimes with the 
eye only, how many of the rectangles are con- 
tained in the map, and consequently, how many 
acres. When the map is laid down by a scale of 
20 perches to an inch, then each rectangle will be 



The Surveyor's Guide. 



55 



16 perches by 10, or one acre ; and if the map be 
40 perches to an inch, then each rectangle will be 
32 perches by 20, or 4 acres ; and if by 80 per- 
ches to an inch, then each rectan- 
gle will contain 16 acres. This 
instrument would be useful to 
gentlemen and others not very 
well skilled in surveying, to 
measuriB a map, or part of a 
map that they wished to know 
the content of nearly. It is 
easily used. The sides of the ^^s- 19. 

glass must be made to coincide with as many of 
the lines on the map as possible, and the broken 
squares can be estimated by the eye, or a square 
inch horn. 



Description and design of a new instrument by wliicli dis- 
tances can be found at once, without any calculation ; 

Let a brass semi-circle (fig. 26) of about 9 inches 
radius, have its inner edge or limb, divided into 
90 equal parts, beginning at N and counting up- 
wards 10, 20, 30, &c., to 90 at Z, and each of 
these divisions subdivided into 6 equal parts. Let 



56 The Surveyor's Guide. 

the outer limb be divided into degrees and 6th 
parts of a degree, marking the degrees from the 
middle of the limb, both ways, 10, 20, 30, &c., 
to 90 at N and Z. Let also, the middle space be- 
tween the outer and inner limbs, be marked from 
Z to N, 10, 20, 30, 40, &c., to 180 at K 

Let this semi-circle be fixed to the middle of a 
box ruler B D, about S^ feet long, an inch and a 
half broad, and of a convenient thickness. The 
inner breadth of half this rule must be level with 
the surface of the semi-circle, but the outer half 
must be higher about two-tenths of an inch. On 
the outer half there must be fixed a thin brass 
scale of an equal length and breadth with the box 
ruler, the breadth of which scale is to be divided, 
by lines drawn from end to end, into three equal 
parts, and the length into inches, half inches, and 
tenth of an inch ; the inches are to be drawn di- 
rectly across the whole breadth, and marked 1, 2, 
3, 4, &c., both ways to B and D ; the half inches 
are to be drawn across the middle and innermost 
third, and the lOths only across the inner third. 
Let there be on one end of this scale an inch, and 
on the other end half an inch, each divided very 



The Surveyor's Guide. 57 

exactly into 10 equal parts diagonally, that the 
lOths and centesms which may happen in the 
operations, on the square and indices hereafter to 
be described, may be exactly measured on them by 
a pair of dividers. The reason for raising the 
outer half of the box ruler above the inner half 
two-tenths of an inch, is to make room for the in- 
dices A b and A d, which are to be fixed to the 
centre of the semi-circle, and there to open and 
shut as occasion requires, like the legs of a sector. 
Those indices are about 26 inches long, three- 
fourths of an inch broad, and about two-tenths 
thick; their breadth is to be divided into three 
equal parts, and their length into inches, half 
inches, and tenths, as the brass scale before men- 
tioned. The inches are to be marked from the 
center A, with 1, 2, 3, 4, &c., to b and d, and the 
tenths drawn across the inner third. Each of 
those indices must have a small screw nut with a 
pin or bit of wire upon it, which pin may, by the 
screw nut, be fixed exactly to any division on them 
in order to suspend the label, or ruler T Y, which 
has a thin piece of brass with a small hole in it, 
exactly fitting the aforesaid pin, and is to be fixed 



58 The Sueveyor's Guide. 

also to any division of the ruler, as occasion re- 
quires. Let this label, or ruler, be about two feet 
long, and of the same breadth and thickness as the 
indices A b and A d, and divided after the same 
manner as they are, only the tenths are to be 
drawn across the inner edge, as well as across the 
inner third of the breadth, and the inches are to 
be marked 1, 2, 3, 4, &c., from C to T and Y, 
making C T eighteen inches, and C Y six. The 
like divisions are to be made on the side of the 
square K X, beginning at the inner edge of the 
brass ruler at K, marking the full inches on the 
upper side, 1, 2, 3, 4, &c., to 24 ; the tenths are 
to be drawn across the upper third and the upper 
edge. Let this instrument be fixed on a tripod 
with a ball and socket like those of a common sur- 
veying instrument, but very strong, in order to 
have it very firm ; and let there be sights which 
may, as occasion requires, be fixed on the diame- 
ter, indices, and ruler T Y, of the the same kind 
with those of a surveying instrument. 

N. B. The ball and socket must not be fixed 
exactly under the center of the semi-circle, but some 
distance from it, on the cross-bar which goes from 



The Surveyor's Guide. 



59 



the center to the middle of the limb, as -well to sup- 
port the head of the instrument more easily by be- 
ing nearer its center of gravity, as to make room 
for an air level, which must be fixed exactly under 
the diameter or ruler A B, so that when the semi- 
circle is turned vertically the diameter may be 
fixed horizontally. 




60 The Surveyor's Guide. 

The use of the Instrument in measuring distances : 
Example. 

Let it be required to find the distance from the 
house at A to the castle, (fig. 27) or to any part 
thereof, as the weather-cock on the top of the 
spire at C. 

Having set up your instrument at A, turn it 
about till through the sights on the diameter, you 
see a mark set up at B, and having fixed the di- 
ameter in that position, turn the moving index till 
through the narrow slit of a small sight fixed on 
the center, you see the hair in the other sight cut 
the spire at C, then fixing the index in that posi- 
tion to the limb of the semi-circle, measure with a 
four pole chain in a straight line from A to B ; 
and having marked the chains and links of that 
distance on the diameter and placed the ruler with 
the sights on it exactly to that distance, by means 
of the small pin and hole mentioned before, set up 
your instrument at the end of the distance you 
measured (which you may make full chains if you 
please) and turn it about till through the sights on 
the diameter you see a pole at the first station A, 
and having fixed it in that position, turn the ruler 
on the pin which is fixed at the former distance on 



The Surveyor's Guide. 61 

the diameter, till through the sights on it you see 
the vane at C ; then will the part of the index a c, 
cut by the inner edge of the ruler, give the dis- 
tance A C from the house to the spire at C, which 
was to be found ; and if there be occasion, the dis- 
tance from the mark at B to the spire will be found 
on the ruler at the intersection of the index ; all 
of which is plain from the similarity of the trian- 
gles ABC and a. pin. c, or that formed by the 
diameter, index, and ruler, from Cor. 1st 4 Euc. 
Book 6th. Thus the surveyor can find the distance 
of any or all the particular objects he can see and 
may wish to set down in his map, and by turning 
the instrument vertically by means of a notch in 
the socket, inaccessible heigths can, in like manner, 
be readily ascertained in the same manner. 

Example in Measuring Distance. 

Let it be required to find the distance from the 
house at A to the castle, (fig. 27) or to any part 
thereof, as the weather-cock at the top of the spire 
at C. 

Having set up your instrument at A, turn it 
about till through the sights on the diameter, you 
see a pole at B, and having fixed the diameter in 



62 



The Surveyor's Guide. 



that position, turn the moving index till through 
the narrow slit of a small sight fixed on the center, 
you see the hair in the other sight cut the spire at 
C ; then fix the index in that position to the limb 
of the semi-circle and measure with your chain of 
100 links in a straight line from A to B, which 
mark on the diameter, and place the ruler, having 
the sights on it exactly on that distance by means 
of the small pin and hole before mentioned ; set 
up the instrument at the end of the measured dis- 
tance, and turn it about till through the sights on 




Pig. 27. 

the diameter you bisect the pole at A, and having 
fixed it in that position, turn the ruler on the pin 
which is fixed at the former distance on the diame- 
ter, till through the sights you see the vane at C ; 
then will the part of the index, a c, cut by the in- 



The Surveyor's Guide. 63 

ner edge of the ruler, give the distance A C from 
the house to the spire at C. 

And in like manner by directing the ruler to 
any other objects from A, and noting the degrees 
cut by the ruler on the limb, and directing from 
B to each object, the distance from A vrillbe shown 
as before explained, and thus the surveyor fur- 
nished with such an instrument, can from the end 
of his first station, tell the length of his diagonals 
to as many corners as he can see from that point. 
Also, by turning the instrument vertically, heights 
can be determined in the same manner. 

I would recommend the surveyor to use a com- 
pass, having the limb divided into 360°, and the 
bottom of the box into four 90's ; then in taking 
the courses, if N. W., the limb and quarter com- 
pass are the same ; but if in the S. W. quarter, 
the sum of the degrees on the limb and quarter 
compass are 180° ; and in S. E. quarter, the dif- 
ference of the degrees on the limb and quarter 
compass make 180° ; lastly, if in the N. E. quar- 
ter, the sum of the quarter compass and limb 
make 360. A surveyor should prove all his courses 
by this rule before he quits his instrument. 



64 The Sueveyor's Guide. 

Problem. 
Given the bearings of any two stations of a 
survey, thence to determine the angle made by 
those stations. Rule — Deduct the preceding bear- 
ing from the succeeding, according as the remain- 
der is greater or less than 180°. Add — or+180° 
(as the case may be) and you have the required 
angle. 

N. B. The angle found by the above rule will 
be internal if the polygon lie towards the right 
hand in the traverse ; and external, if toward the 

left. 

Example First. 
Required the several angles of the polygon 
A B C D E F G, the courses of the sides being, viz : 




Kg. 28. 



1 


A B 2691° 


or 


S. E. 


891 


2 


B C 2511 


or 


S. E. 


711 


3 


C D 252f 


or 


S. E. 


72| 


4 


D E 162^ 


or 


s. w. 


in 


5 


E F 77| 


or 


N.W. 


77f 


6 


FG 30| 


or 


N.W. 


30^ 


7 


GA 5f 


or 


N.W. 


5f 



The Surveyor's Guide. 65 

From 2511 From 77| 

take 269J take 162^ 

—18 —841 

+180 +180 

Sum 162=Ang. ABC. Sum 95i=Z. DBF 

From 252| From 30| 

take 2511 take 77f 

+li —47 

+180 +180 

Sum 1811=/. BCD. Sum 133=/: E F G 

From 162i From 5| 

take 252| take 30| 

—901 _25 

+180 +180 

Sum 891= A C D E. Sum 155=/. F G A 

From 2691 
take 5| 

Rem. 2631 
—180 

Sum 

Now 180° multiplied by the number of sides 
6* 



66 The Surveyor's Guide. 

any polygon minus 360°, equals the sum of the in- 
ternal angles .-. 180X7= 160 and 1260—360=900 
So 83|+162+181i+89J+95i+133+155=900°. 
Proof. 

Next. Having the bearing of any station and 
all the internal angles of any polygon, thence to 
determine the courses of each of the other stations 
in the regular order of succession, viz : the land 
lying to the right hand as you surround it. Rule : 
According as the given angle is+or — than 180° ; 
add the preceding bearing, succeeding angle, and 
+or — 180° (as the case may be ;) their sum will be 
the succeeding bearing or course. 

Note. — It sometimes happens that the result 
will be more than 360° ; in this case take 360° 
from it and the remainder will be the course of the 
succeeding station. 

Example. 

Take the course of A B 269i or S. 89i E, in the 
preceding figure, and the angles as there found, 
viz : 



The Surveyor's Gcjide. 67 

269^ 162i 

162 95J 

+180 +180 ^ 

611J 4371 

Deduct 360 360 

Cou. of B C 251| or S TIJ E. Cou. of E F 77| or N 77f "W 

251i 77f 

ISli 133 

--180 +180 

Cou. of C D 252f or S 72$ E. 390f 



252f 



360 



fj Cou. of F G 30| or N 30| W 



+180 

30| 

522i 155 

Deduct 360 +180 

Cou. of D E 162i or S 17| W. 365f 

360 

Cou. of G A 5forN.5|W. 

5f 
83f 
+180 

Cou. of A B 269 J or S 89 J E. 

being the' same as that given ; therefore, a proof 
of the correctness of the work. And thus the sur- 
veyor has a sure method of avoiding the inconve- 
nience of the needle being drawn from its true 
position by mines or other causes, and also correct 
the diurnal variation ; for no matter how much 



68 The Surveyor's Guide. 

the needle may be attracted at any station, the 
angle will be correct by taking a back and fore 
sight at every station, and having the true course 
of the first station. All the others can be found 
by the foregoing rules. And to know if any at- 
traction exists at the first station, take a course in 
a different direction from your chain line ; go to 
the object bisected, or to some convenient distance 
in that direction, and take a back sight ; if that 
agree with the fore sight, you may safely conclude 
that no attraction exists at either ; but should it 
differ, make trial in some other direction, in like 
manner, till you find what station the attraction is 
in ; but by using a good theodolite all such trouble 
is avoided. 

In every survey that is truly taken, the sum of 
the Northings is equal to the sum of the Southings, 
and the sum of the Eastings to the sum of the 
"Westings. 

Let a b c e f g h represent a plot or parcel of 
land ; let a be the first station, b the second, c the 
third, and so on. Let N S be a meridian line, then 
will all lines parallel thereto, which pass through the 



The Surveyor's Guide. 



69 




y c 4 

several stations, be aj 
meridians also, as a o, 
b s, c d, &c., and the 
lines b 0, c s, d e, 
&c., perpendiculars 
to these, will be east 
or west lines or de- n"T 
parture. The northings e i+g o+h g=a o+b s+ 
c d+f r, the southings ; for let the figure be com- 
pleted, then it is plain that g o+h g+r k=a o+ 
b s+c d and e i — r k=f r; if to the former part 
of this first equation e i— r k, be added, and f r 
to the latter, then g o+h g+e i=a o+b s+c d+f r ; 
that is, the sum of the northing is equal to the sum 
of the southings. 

The eastings c s+q a=o b+d e+i f+r g+o h, 
the westings for a q+y o (a z)=d e+i f+r g+o h, 
and h o=c s — y o. If to the former part of this 
first equation c s — y o, be added, and b o to the 
latter, then c s+a q=o h+d e+i f+r g+o h ; that 
is, the sum of the eastings is equal to the sum of 
the westings. 

Now, as there is many methods of calculation, 
and every man chooses one in preference to all 



70 



The Surveyor's Guide. 



others, I shall here show the method which I have 
always practiced, being, I think, least liable to 
mistakes, although not the shortest, as shall be 
hereafter shown. 



1^ 






: : o : : : 


1.00 
2.00 
2.00 
2.00 
1.00 


s 




IJjj igllJ i 


: : : 


i 


^ 


O 
-* 




iii 1 1! 


ilil 




i 






: : : 




11 


ogooooooooooooo 
















g|§|§ :|g| :J : 


o 


sSsiijiiS 




lis 


o 


k 


11^3333333331331 


1 






"Si 


rH « «) ■* >0 -iO i^- 00 OS C 


r-l <N M 'il i« 




















o o 
3 I 

CQOQ 



In the above method the northings and south- 
ings, eastings and westings, being corrected by the 



The Surveyor's Guide. 71 

foregoing rules, set the sum of the northings, or 
southings at the top of the column titled latitude, 
then continually add the northings and subtract 
the southings, or add the southings and subtract 
the northings, and the last number will always be 
the same as the first, which is a proof of so much 
of the work. Then add the first and last latitudes 
together, and place their sum opposite to the first 
station in the column under latitudes, added, and so 
continue to add every two adjoining latitudes, and 
place their sum in a line with the latter, then mul- 
tiply each of these numbers by the particular 
easting or westing belonging to that station, and 
place the product in the column of east or west 
area, as the case may be, and the difference of 
these two columns divided by two, will be the con- 
tent of the survey. In this method there is no 
danger of making mistakes from indirect stations, 
and by using the eastings, and westings, in the same 
manner as you did the northings, and the south- 
ings, you can prove the work, and find the area 
four different ways. 



72 



Surveyor's Guide. 



ANOTHER METHOD, WHEREIN FEWER FIGURES ARE 
USED, NEVER BEFORE PUBLISHED : 

The Eastings and Westings, Northings and South- 
ings, are here corrected according to the foregoing 
rules, and placed as usual, as follows : 

CALCULATION OF THE NOTES ON THE SUCCEEDING 
PAGE. 



1 1 1 1 Lata. IDoub. Semi 1 1 1 
N. 1 S. |li. N.|L.S. 1 added. 1 Eectangle.l E. 1 W. 1 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 

I 




2.76 
4.44 
0.77 
4.38 


2.76 
7.20 


5.15 
4.38 
0.00 
2.60 
2.96 
6.31 

...... 


2.76 
9.96 
9.53 
4.38 
2.60 
6.56 
9.27 
8.74 
3.29 
1.58 
4.51 
5.29 
1.77 


+4.4712 
+11.9520 
+ 20.5848 
— 1.9272 
14.3520 
—10.3416 


1.62 
1.20 

0.44 

1*86 


Bx.E 
2.16 

'6.38 
'l.34 

15.40 












Ex. S. 
2.60 
0.36 
3.35 
3.34 
2.11 


Indirect. 










6.04 
2.70 
0.59 
0.99 
3.52 
1.77 
0.00 


+59.1426 
+ 17 4800 


Ex.W 
9 nn 




'"o.'io 

2.53 


—4.4086 

+5.6564 3.58 
+5.4120 1.20 


Indirect 


1.75 
1.77 


+5.0784 
+4.4958 


0.96 
2.54 

15.40 




Ex. N. 




15.28 
)ouble t 


15.28 
le sum 


)f th 


aindi 


rect, 


165.3026 
33.3548 






131.9478 





65.9739 Angular.spaces. 



The Surveyor's Guide. 



73 



11.76 Parallel breadth. 
12.35 Meridianal breadth. 



5880 
3528 
2352 
1176 

145.2360 Content of 
65.9739 


parallelogram. 


7,9.2621 
4 




3.70484 
40 




28.19360 

A.R. P. 

7.3.28.19, the same as on nest page. 

Nl 




74 



The Surveyor's Guide. 



The foregoing plot and calculation may not be 
unacceptable to the reader, being as complicated a 
figure as could be easily met with. 

A new and concise method of Calculation, wherein fewer 
figures are used than in the common methods : 



1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 


"i'eo 

0.36 
3.35 
3.34 
2.11 

"l."75 
1.77 


S. 

2.76 
4.44 
0.77 
4.38 


E. 

1.62 

1.20 

"6.44 


W. 


M. D. 
162 


D. D. 


Area. 


Deduction. 


"2.I6 

"s.'si 
"e.'ss 
"i'.u 


3.24 E. 
4.44 E. 
2.28 E. 
2.72 E. 
2.80 W. 
0.94 W. 
7.32 W. 
5.32 W. 
6.66 W. 
3.08 W. 
1.88 W. 
0.92 W. 
1 62 E 


4.86 E. 

7.68 E. 

6.72 E. 

5.00 E. 

0.08 W. 

3.74 W. 

8.26 W. 
12.64 W. 
11.98 W. 

9.74 W. 

4.96 W. 

2.80 W. 

0.70 E. 


13.4136 
34.0992 

5.1744 

21.9000 

2080 

1.3464 
27.6710 
42.2176 
25.2778 














1.86 






"o!40 
2.53 


2.00 

"■3.58 
1.20 
0.96 
2.54 






3.8960 

12.5488 




4.9000 




1.2390 









15.28 15.28 15.40 15.40 



17.6838 



A. R. r. 

7 3 28.19 



2,0)15,8.5242 

7,9.2621 
4 



3,7.0484 
40 



This method may be called a compound of 
Burgh's and Gibson's, without being intimately 
connected with either. It allows the first meridian 
to pass at any distance from the first station not 
less than the first latitude or first departure. 



The Surveyor's Guide. 75 

This example supposes the first meridian to pass 
at the distance of the first Easting from the first 
station of the survey, and the M. D. column is 
completed by one single addition of the Eastings, 
or one single subtraction of the Westings, to or 
from each preceding one, agreeably to the nature 
of the signs. The D. D., or double distance col- 
umn, is completed by adding the first and last, 
and placing their sum in a line with the first East- 
ing or Westing, and then adding every two ac- 
cording to the signs, and placing their sum in a 
line with the latter, marking E. or W. as the case 
may be. Then the Eastings X by the Southings, 
and the Westings X by the Northings, must be put 
into the area column ; but, the Westings X by the 
Southings, and the Eastings X by the Northings, 
must be put into the deduction column, the diflfer- 
ence is double the area of the survey. 

The following is a method of calculation first 
published by Noble, the inventor, and is a very 
superior plan when well understood, but requires 
considerable attention to distinguish the indirect 
stations, as the areas belonging to them must be 
deducted. A little practice will enable the learner 
to know both them and the four extremes, viz : 



76 The Surveyor's Guide. 

N. S. E. and W. That author's description of a 
semi-rectangle is a figure limited by the latitudes 
of both ends of the station, the station itself, and 
a section of the parallel from which the latitudes 
are measured, equal to the departure ; and when 
the last mentioned is indirect, the semi-rectangle 
is indirect also, viz : Indirect or retrograde sta- 
tions are those stations, in respect of the rest, 
which bear backward or contrary to the natural 
succession of the four quarters of the compass. 

If, in proceeding Southerly from the extreme 
point jSTorth, there happen a station to turn North- 
erly, or, in proceeding Northerly from the extreme 
point South, there happen a station to turn South- 
erly, such stations are indirect or retrograde sta- 
tions. The same may be said of stations that 
turn after the like manner in proceeding from the 
extreme points E. and W. of the survey. The 
extreme points, N. S. E. or W. of a survey, are 
the ends of those stations which run more to the 
N. S. E. or W. than any other stations in the survey. 

Though most surveys have those four extreme 
points, yet there are some where one and the same 
station may be the greatest extreme N., and at 
the same time the greatest extreme East or West ; 



The Surveyor's Guide. 77 

or one and the same station may be the extreme 
South, and likewise the extreme East or West. 
The circumscribing parallelogram of a survey is a 
rectangle or parallelogram circumscribing the body 
of the land, whose four sides, passing through the 
four extremes N. S. E. and W. of the survey, are 
two meridians and two parallels of latitude. 

The angular spaces are the areas contained 
between the sides of the circumscribing paralello- 
gram, and the stations of the land surrounded, 
which, deducted from the area of the second parall- 
elogram, leaves the content of the survey. 

Now in order to find the area of those angular 
spaces, the four extremes must first be ascertained. 
This an experienced hand can see at once by 
examining his field-book, which, being known, you 
must find the latitude of each station in the survey 
from the extreme points North and South ; thus, 
having found and corrected your latitudes and 
departures, and placed them as in the following 
table, write in a line with N, and also the South 
extreme as in the following table. Now, begin- 
ning at each of these extremes. North and South, 
continue to add the Northings, and subtract the 
Southings to find the latitude of each station to 
7* 



78 The Surveyor's Guide. 

the extreme point West, but you must still add 
the Southings and subtract the Northings to the 
extreme point East. When the latitude of every 
station is thus found, and placed in their proper 
columns, add every two latitudes next each other, 
and put their sum in a line with the latter station 
in the column marked L. A., and each sum or 
number in this column is the length of a rectangle, 
which is double the semi-rectangle of each station. 
It is no matter at which of the two latitudes you 
begin, so that you place their sum in a line with 
the latter or succeeding station ; but it is common 
to begin by adding the first and last' stations to- 
gether, and placing their sum in a line with the 
first station ; then add the first and second, and 
place it in a line with the second, and so on 
till the column is filled. Then each number must 
be multiplied by its corresponding Easting or 
Westing, and the products put in the column 
marked D. S., or double semi-rectangle of each 
station. If the Easting or Westing be direct, 
then this product must be marked + ; but if it be 
indirect, with the negative sign — , and the sum 
of all the affirmatives, abating the sum of all the 
negatives, will be the content of all the angular 



The Surveyor's Guide. 



79 



spaces. But, to find tlie length and breadth of 
the circumscribing parallelogram, note that from 
the sum of all the Northings or Southings you 
must deduct the sum of all the Northings or South- 
ings that have indirect diiference of latitude, 
•which will give one side, and the same must be 
done with the Eastings and Westings to find the 
other side. The length and breadth of the parall- 
elogram being thus found, they must be multiplied 
together, and from their product take the content 
of the angular spaces, and the remainder will be 
the content of the survey. 

TAKE THE FOLLOWING EXAMPLE IN NUMBERS. 





N. 


S. 


L.N. 


L.S. 


1 




5.75 




0.00 


? 


30.28 

9n 






30.28 


3 






m 51 


4 


9.04 




6.17 


48.55 


5 


4.66 




1.51 




6 


151 




0.00 








6.78 


6.78 




8 




17.46 


24.24 




9 




12.97 


37.21 




10 




11.76 


48.97 


6.75 



5.75 

30.28 

69.79 

88.06 

7.68 

1.51 

6.78 

31.02 

61.45 

86.18 



-076.5900 

-468.7344 

-687.4315 

-626.1066 

+088.9344 

+020.2340 

+049.8330 

+268.3230 

— 233.5100 

+739.4244 



11.58 
13.40 
7.35 



.58 



13.32 

15.48 

9.85 

7.11 



2)2792.1013 49.56 49.56 



1396.0506=Ang. spaces. 

In this example there are no indirect stations 
in the Northings or Southings, 54.72 is the me- 
ridianal breadth of the survey. But station 9th 
being indirect in the parallel breadth, must be 



80 The Surveyor's Guide. 

deducted from the sum of the Easting or Westing 
to find the other side of the circumscribing par- 
allelogram. Thus : 

49.56 Sum E. or W. 
3.80 Indirect. 



45.76=Parallel breadth. 
54.72=Meridianal breadth. 



91.52 
32032 
18304 

22880 

2503.9872 Content of circum. parallelogram. 
1396.0506 " of the angular spaces. 

1107.9366 
4 

3.17464 
40 



6.98560 110.3.06.98, the content. 

In this example you may see that the four ex- 
tremes are the 6th, 1st, 10th, and 4th stations. 
You can also see that the two latitudes of the 
extreme West is equal to the two latitudes of the 
extreme East, that is 6.17+48.55=48.97+5.75, 
which is a proof to so much of the work. 



The Surveyor's Guide. 



81 



If you begin with the Eastings and Westings, 
and proceed as you were directed, all along with 
the Northings and Southings, you can find the 
content of the survey in like manner, and so prove 
the work. 

To survey with the compass througli any mine, or other 
cause for drawing the compass needle off its parallelism : 

The diurnal variation of the needle is known 
to every practical surveyor, but is easily cor- 
rected by examining the time of the day when the 
courses of long stations were taken ; as from about 
8 o'clock in the morning till about 2 in the after- 
noon, the needle varies Westerly to from about 
7'08" to about 13'2r', as shown in the following 
table. The surveyor can make such allowance as 
will (all other errors apart) insure a complete 
clase. 

MEAN DIURNAL VARIATION FOR EVERY MONTH IN 
THE YEAR. 



January, 


0" 7'08" 


July, 


0"13'14" 


February, 


8'58" 


August, 


12'19" 


March, 


ivn" 


September, 


11'43" 


April, 


12'26" 


October, 


10'36" 


May,' 


13'00" 


November, 


8'09" 


June, 


13'2F 


December, 


6'58" 



82 The Surveyor's Guide. 

Now, in surveying with the compass detatched 
from a Theodolite, both back and fore sights 
should always be taken ; and to make sure that no 
attraction exists in the first station, take a course 
in a contrary direction to some object, go to that 
object and take a back sight ; if the fore and back 
sight agree you may be satisfied that no attraction 
is at your first station ; but should they not agree, 
you must then, from the latter station, repeat the 
like process till you find at which of them the 
attraction exists ; if, at the first station, either 
note its quantity, which allow on the next course, 
as in tracing old boundaries ; or pay no attention 
to it at the starting, but continue to take the fore 
and back sights throughout, and as at any station 
the needle will be as much attracted at the fore as 
the back sight, the angles can all be truly found as 
formerly shown, and thence the true courses for 
calculation by latitude and departure. Thus may 
the expert surveyor traverse any city, mountain, or 
other place containing mines or other substances 
which attract the needle, about which I have heard 
many complaints. 



The Surveyor's Guide. 



83 



Now to plot the last given notes, and in like 
manner any other survey similarly prepared : — 
Having the length and breadth of the circumscri- 
bing parallelogram, let it be drawn by the same 




scale you intend to lay down' your map by, and 
beginning at either of the extremes, as 1, lay off 
your latitude as l.a 5.575, and at right angles to 
that, the departure of that station or Westing a.2 
13.32, and join their extremities with the line 1 2, 



84 The Surveyor's Guide. 

which is the distance. The next station is N. W. 
Draw toward the North 2.b parallel to the sides 
of your parallelogram, and on it lay 30.28, your 
next Northing, and at right angles thereto toward 
the West 15.48, your next Westing, and join 2 and 
3, which is your next distance, and so on all round, 
and as your Northings are equal to your Southings, 
and your Eastings to your Westings, your last 
departure, whether East or West, will fall into the 
point of beginning, as T.l. This is the most expe- 
ditious mode of plotting surveys, and can be made 
use of in the most extensive work, and is much 
superior to protraction by parallels and a metallic 
protractor. The mechanical methods of finding 
area, shown by many authors, I do not think well 
to notice, as none of them can be depended on for 
accuracy. 

OF LOTTING OR LAYING OUT TOWNS, &C. 

Regarding this kind of surveying, little can be 
said more than giving some general directions con- 
cerning the method of operation, as every man has 
mostly predetermined the manner in which he 
intends to have his property cut up into lots. 
Provide yourself with a 20 or 25 feet pole, ten 



The Surveyor's Guide, 85 

skivers with sharp points and thin edges, two brass 
plummets with steel points hung to fine cords ; 
then having fixed poles so as to direct you in a 
straight line, and set them perpendicular by the 
help of your plumb, direct your assistant to hold 
one end of your pole in the straight course, with 
his plummet hanging over the extremity, whilst 
you hold yours touching the end of the pole which 
you hold, and the point of your plummet exactly 
over the starting point ; when both plummets are 
steady, order your assistant to stick, and exactly 
where the point sticks, he sticks one of his skivers 
edgewise and slanting, so as that you can, when 
you arrive at it, hang the point of your plummet 
exactly over the edge of the skiver, and your 
assistant again sticks his plummet in the ground, 
and a skiver as before, and so on to the end. By 
measuring carefully in this manner, property can 
be laid out with great accuracy. 

Almost every man has his own method of keep- 
ing his field-book, but the following method, which 
I have always adopted, is, I think, best calculated 
to prevent confusion in extensive surveys, for as 
writing backward and chaining forward are con- 
trary, it is more congenial, and natural, to both 
8 



86 



The Surveyor's Guide. 



write and chain forward, by beginning at the bot- 
tom of the page. 

N. B. It may not be unacceptable to the reader 
to see these notes, calculated by Noble's method, 
as on page 47. 



The Survetor's Guide. 



87 



Maple 


® 

60.00 
35.10 

N.77fW. 


Tin Oak 
0-5 








W. 0. 


(5) 












Stump. 


«i«, 










1 


41.40 










162i 










"a 


S.17IW. 










1 
1© 


(4) 


To a stone. 


i 

1 

3 




To the place of 


m 

36.40 


18.48 


"^ 0.6 


33.00 








heginning. 




© 




1 


@ 






21.00 




7.10 


R. 0. 




S 


S. 0. 




3.24 


0.15 to a Pine 








N. 5f E. 


@ 




15.00 


0.10 




(7) 




7.00 
2521 

S. 72JE. 


toChesiiut 










^ 




J^ 




1 


(3) 


To a post. 


1 


36.25 
23.00 


To a post. 
Chesnut. 


B.Oali 


41.00 


s 


26.00 
4.00 


Dogwood 


1 

n 




1 


2511 

S. 71|E. 

(2) 

2olo 
26U 


chains to a 
hickory 


10.00 

7.00 

N.30fW. 

(6) 


0.10 to a Beech 

Shellbark 
mckory. 


m 




S.89|E. 


tree. 




66.00 












Stream North 




(1) 






62.00 


36 West. 



Begins at a White Oak on. Squire Hays' Estate. 



S8 The Surveyor's Guide. 

The foregoing method of keeping a field book, 
I think, is the most convenient I have seen. The 
following is the calculation of the notes corrected 
by the foregoing rules. 







P. 






Lat. 












1 




Poles. 


N. 


S. 


32.05 


L.A. 


E. Area. 


W. Area. 


E. 


W. 


S.89V 
S.7i4Ie. 


10.00 




0.10 


31.95 


64.00 


640.0000 




10.00 




2 


20.50 




6.55 


25.40 


57.35 


1115.4575 








3S.7a^E. 


18.40 




5.50 


19.90 


45.30 


795.9210 




17.57 




4S.17gw. 


20.90 




19.90 


0.00 


19.90 




126.9620 




6.38 


5 N.77gW. 

6 N.30^Vy. 

7 N. 53| E. 


33.00 


6.9+ 




6.9+ 


6.94 




223.8150 




.32.25 


18.25 


15.fiS 




22.62 


29.56 




275.7948 




9.33 


9.48 


9.03 




32.05 


54.67 


51.3898 


1 0.94 





Double axea, 



626.5718 47.96 47.96 



2.0)1976.1965 in square four pole 
chains. 



A.R. V. 

98.3.0957 







Lat. 


Lat. 


Lats. 


DouWe Semi- 






N. 


S. 


South. 


North. 


Added. 


rectangles. 


E. 


W. 




0.10 




0.10 


0.10 


—1.0000 


10.00 






6.55 




6.65 


6.75 


+131.2S75 


19.45 






5.50 


19.90 


12J5 


18.80 


4-330.3160 


17.57 


Ex.E. 


Ex. S. 


19.90 


00.00 




19.90 


—128.9620 




6.38 


6.94 




6.94 




6.94 


+22:!.S150 




32.25 


15 68 




22.62 


9.43 


29.56 


■f275.7M8 


Ex. W. 


9.33 


9.43 


North. 




. 00.00 


9.43 


+8.8&42 


0.94 . 





32.05 32.05 
Content of the angular spaces, 



The Surveyor's Guide. 

47.96 Parallel breadth. 

32.05 



23980 
9592 

14388 



1537.1180 Content of circum. parallelogram. 
549.0197 " of the angular spaces. 

988.0983 
4 



3.23932 
40 

9,57280 



A. R. P. 

98.3.95.73 the same as before. 



There are no indirect stations in the above, but 
were the longitudes made use of instead of the 
latitudes, the last station would be indirect; and 
here also it may be seen that the sum of the oppo- 
site latitudes, against the extremes East and West, 
are equal, viz : 12.15+19.90=32.05 and 22.62+ 
9.43=32.05. 

Of the Tracing of Old Hearings. 

Gummer, in his work on Surveying, gives the 
general number 57.3°, for doing this which many 
8* 



90 The Surveyor's Guide. 

work with, although it is not correct, but comes 
out pretty near the truth when the chain line is not 
very long. 

To find this number, say 6.2831853 (the circum- 
ference of a circle whose diameter is 2) : 360° :: 1 
: 57.3° nearly. Now if two corners are known, 
and can be both seen, set your compass at one of 
them, and direct your sights to the other ; the dif- 
ference between that shown by your needle, and 
that shown in the deed, will be the variation to be 
allowed on each course round the land, supposing 
all those given in the deed to have been correctly 
taken at the time the survey was made, which fre- 
quently happens not to be the case. If the two 
corners cannot be seen from each other, run the 
course and distance given in the deed, and observe 
if the point you arrive at, joined to the corner, 
form an Isoseless triangle, which will be the case 
if all be right ; otherwise some mistake has been 
made in the distances, which must be corrected. 
Then take the pendicular distance to the given 
corner, and say : As the measured distance is to 
the distance to the corner, so are 57.3° to the 
number of degrees, minutes, or seconds, as the 
case may be, which will be the variation. Or, 



The Sukveyor's Guide. 91 

more accurately. As the distance to where the 
perpendicular was taken is to radius, so is the dis- 
tance to the corner to the tangent of the variation. 
In running your trial line, you will be told you are 
wrong, and that you don't understand your busi- 
ness, and all such stuff, will be sounded in your 
ears ; but pay no attention to such nonsense, for it 
is to be regretted that too many men are so igno- 
rant as to think that a Surveyor can, by some 
mysterious means, direct his compass on the exact 
line, and find all the courses as if by magic. It 
often happens that the corners runs through clumps 
of trees or other obstructions through which you 
cannot chain. In such a case I have often chosen 
an opening some degrees to right or left of the 
fence, and at certain distances driven posts till I 
found a perpendicular to the corner. Then, as 
the whole distance is to the perpendicular, so is 
each distance from the beginning to the perpen- 
dicular distance from the measured line to the 
fence, which, being correctly laid ofi", and posts 
driven at their extremities, will point out the true 
boundary. 

Of Levelling. 
The art of levelling consists in finding or tra- 



92 



The Surveyor's Guide. 



cing a line on a given portion of the earth's sur- 
face, parallel to the horizon at all points. The 
subject is too extensive to be comprised in this 
small treatise. I shall give an example, which it 
is hoped will enable the reader to do anything of 
that nature that may come in his way. Any one 
desirous of being fully informed on that subject, 
should consult Bruff's Engineering, where every 
information on that subject can be obtained. Re- 
garding the adjustment of the level, which is a 
simple matter, let the practitioner always place his 
level in the middle, between the back and fore- 
sights, and keep the bubble in the middle of the 
divisions, and all will be right. 



The Surveyoe's Guide. 



93 



FIELD BOOK DISTANCES, MEASURED WITH A ONE 
nUNDRED FEET CHAIN. 



Efleva- 
tion. 


Back 
sight. 


Fore 
sight. 


Depres- 
sion. 


Total 
elevat'Q 
Datum 
100 feet. 


Dis- 
tance. 


...... 


1.11 


6.S4 
5.73 
8.10 
8.15 
5.80 
5.00 
5.01 
4.05 
4.98 
6.12 
2.25 
7.77 
3.95 
6.30 
1.60 
4.24 
6.74 
2.17 
4.60 
5.36 


5.73 
8.10 
8.15 
5.80 
5.00 
4.55 
4.05 
4.98 
6.12 
6.67 
7.77 

13.52 
6.30 

10.80 
4.24 
6.74 

10.20 
4.60 
5.36 
5.99 




100. 
101.11 

98.74 
98.69 
101.04 
101.84 
102.29 
103.25 
102.32 
101.18 
100.63 
95.11 
89.36 
87.01 
82.51 
79.87 
77.37 
73.91 
71.48 
70.72 
70.09 


.00 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 

100 


] On cross road 50 
J ft. W. N. 54 W. 
1 Side of ravine, 30 
J feet deep. 
10. bottom 10 ft.wd. 
1 Top of bank on 
J other side. 

1 Middle of stream, 
i N. 57 W. 


2.37 
0.05 




2.35 
0.80 
0.45 
0.96 








0.93 
1.14 
.55 
5.52 
5.75 
2.35 
4.50 
2.64 
2.50 
3.46 
2.43 
0.76 
0.63 




























5.67 


104.76 


134.67 
104.76 


35.58 
5.67 


100.... 
70.09 


1 Dati 
[therec 
r giving 
J accura 


im at top, from wHoh. 
uoed level is deducted, 
a third proof of the 


29.91 


29.91 


29.91 


cy of the work. 



The fall in the following section fi'om 1 to 21 
is 29.91 feet; this divided into 2100 feet, the 
whole distance gives 1 in 70.21, the regular grade ; 
and to find the grade in degrees, it will be as 2100 
is to radius :: 29.91 to the tangent of the angle in 
this case 0° 49' nearly. Here it will be observed 



94 



The Surveyor's Guide. 



that the difference between the datum line and any 
grade, is th.e height above or below the base line, 
running through the first station. If the ordinate 
be greater, the difference is above base ; if less, 
below. Some old fashioned levellers follow a more 
intricate plan. Thus 101.11—98.74=2.37—1.11 
=1.26 above; again, 98.74—98.69=0.05, and 
1.26+0.05=131 below, and so on. But this re- 
quires too much thought, when to add and when 
to subtract ; whereas the other method is done by 
one subtraction. 




Scale of length, 800 feet to 



To LAY OUT A HOAD ON A REGULAR GrADE UP 

A Hill. 
Set your instrument at the starting point, level 
it, and set the vane on your levelling rod to the 
exict height of the centre of your glass. Ele- 



The Surveyor's Guide, 



95 



vate your grading instrument to the number of 
degrees you intend your road to be. Send forward 
your rod to any place where the cross wire will cut 
the middle of the vane, and there drive a post, and 
on it mark grade, and so on to the end of the 
road. And to find the cuttings and fillings, the 
follotving plan is the most convenient. Set your 
instrument on the starting point, measure very 
exactly the height of the centre of the glass, and 
send your rod to the first point where cutting or 
filling is required. Elevate your instrument to 
the grade, mark where it cuts the rod, and the dif- 
ference of the height of the instrument and height 
on the* rod, will be the cutting or filling. If the 
height of the instrument exceeds that on the rod, 
the difierence is cutting, and per contra. 

Example : 




In the above cut the height of instrument is 4 
feet ; height of rod, 2 ; diiference, 2 cutting. 



96 Surveyor's Guide. 

Again, height of instrument, 4, and back sight to 
rod 5 ; difference, 1 to be added to last, gives 3 
of cutting at instrument. Fore sight, 3 ; differ- 
ence to be added to last, gives 4 feet cutting at the 
rod ; but now, height of instrument, 5 ; back sight, 
2 ; difference, 3 ; which, deducted from 4, leaves 
1, and so through the whole. 

To Inflect in Curves on Rail Roads and 
Others. 

The curves most in use at the present time, are 
those of a circle. The angle made at the angular 
point of the tangents is always given — the length 
of your tangent is also given. To find the radius, 
multiply the natural tangent of half the contained 
angle by the length of the tangent of your curve, 
and the product will be the radius of the curve. 

To find the degree of curvature, divide half the 
chord to be inflected by the radius of the curve, 
and it gives the natural sine of the degrees of cur- 
vature. 

Thus, in the annexed figure, where the radius is 
140, and the cord to be inflected 100. 140)50, 
000000 (.357142 is the natural sine of 20° 55'= 
the degrees of curvature. 



The Surveyor's Guide. 



97 



Demonstration. 

The angle A D C, is a right L=to A D B, and 
A C B is common to the two triangles, ABC and 
A D B. Hence LAC D=B A DJ, the L angle 
of deflection. Now set your instrument at A, 
direct your index to E, turn it towards the curve 

E 




till 20° 55' are told on the limb, holding the end 
of your chain at A; let the assistant hold the 
chain tight, and move round till the other end 
comes in the line of the perpendicular wire of the 



98 The Surveyor's Guide. 

telescope at G, and then fix a pin. Again, if 
nothing intervene to prevent your seeing, inflect 
from A E, double the said /., and fixing one end 
of the chain at G, let the other be stretched to 
come in contact with the telescope at H, and so on 
through the whole. If H cannot be seen from A, 
move the instrument to G, and take a back sight 
to A, and inflect double the L of the degrees of 
curvature from G K, which will fall into H. 

I have met with some calling themselves engi- 
neers, who adopt the following plan. They divide 
57.,3°X60-=3438' by the radius of the circle, mul- 
tiplying the quotient by the number of feet in the 
chord, and divide by 60 for double the angle ; but 
this is erroneous. I remember having met with a 
person who declared that the angle found by this 
rule was the true angle of deflection. I gave him 
the tangent 100, and the radius 100 feet, and he 
did it by this rule, viz: 100)3438(34.38X100= 
3438+60=57° 18'. In this instance the L made 
by the tangent and chord is only 45°, so that in- 
stead of inflecting in 100 feet, this 57° 18' would 
•fall below the chord. Nor is the half of it cor- 
rect, viz: 28° 39'. For by the true method 
100)50(=,5, the natural sine of 30° the true angle. 



The Surveyor's Guide. 99 

It remains to find the length of the curve A G 
H F. The circumference of a circle whose di- 
ameter is 2, is 6.2832 nearly. Hence as 360° : 
6.2882 :: 1 : .01745, &c. This number, multi- 
plied by the degrees in the arc, and by the radius 
of the curve, gives the length of the arc, thus : 

01745 
11 



17450 268.1^' the length of the arc, and 
1745 so of any other. 



1,91950 
140 

7678000 
191950 

268,73000 



The two following problems may be amusing to 
some readers, viz : 

A gentleman has a lot 40 perches long and 30 
perches wide. He thinks the ends may be so ap- 
plied, as that when their extremities are joined, the 
area may be the greatest possible. The perpen- 



100 The Surveyor's Guide. 

dicular breadth, and the length of the unknown 
side are required. 

Answer: Breadth, 26.815 nearly. 
Length of the unknown side, 66.904 nearly. 

Problem Second. 

A plank road is to be made from the city A to 
the town B, 20 miles asunder. A straight road is 
so situated that a perpendicular from A to it is 10 
miles, and from B 6 miles. The plank road must 
touch the straight road in such a point as to be 
the shortest possible by that route, the length of 
the plank road, the point of contact, and radius 
of the curve having 200 feet tangent, are required. 

Answer : Length of the plank road, 25.298 
The distance of the point of contact 

from A, 12.2474 

And from B, 7.3484 

Badius of the curve having 200 feet tangent, 
245 feet nearly. 

This note to be placed after the calculation of 
the large triangle. It is there shown that the area 
of any plane triangle, the three sides of which are 
given, is A B^. Sine B. Sine A 



C 2 Sine C. Which is thus 

proved BAA 



The Surveyor's Guide. 101 

It has been already shown that B C. B A. Sine B. 

2 

equal area of the triangle, .*. B C. A C. Sine C. 
=area. 

2 

Hence B C. B A. Sine B=B C. A C. Sine C. 

Multiply each side by B A, and we have 

B C. B A^. Sine B=B C. A C. B A. Sine C. 

Divide this equation by B C, we have 

B A^. Sine B=A C. B A. Sine C. 

Multiply each side by Sine A .•. 

B A^. Sine B. Sine A=A C. B A. Sine A. Sine C. 

Divide this by Sine C, and 

B A^. Sine B. Sine A. 

■ =A C. B A. Sine A.=twice 

SineC 
the area, and therefore 

A B^. Sine B. Sine A 

=:Area 

2 Sine C 
To find the perpendicular ordinates from the 
chord 6 of any arc of a railroad, in order to set off 
the curve correctly and speedily, without the help 
of an instrument, suppose it to be a 20° curve, the 
tangent 200. Find the radius, as formerly taught ; 
multiply the radius by the natural co. sine of half 
the vertical angle, and you have J the chord. 



102 



The Sukveyor's Guide. 



Multiply the radius by the natural sine of the 
same angle, and you have the distance from the 
centre to the middle of the chord, a constant num- 
ber to be deducted. Now take any distance, sup- 
pose 10 feet, at which you choose to erect your 




ordinates, and from the semi-chord subtract this 
number, square the remainder, and subtract it 
from the square of the radius ; extract the square 
root, from which take the aforesaid constant num- 
ber, and the remainder is the ordinate to be 
rightly applied, and so proceed till you arrive at 



The Surveyor's Guide. 103 

the middle of the chord ; then the difference be- 
tween the said constant number and the radius, is 
the versed sine or greatest ordinate, and now you 
are prepared to lay off the other side of your 
curve, and all this can be done in a few minutes in 
the field. 

Example. See last figure. 

Nat. tangent of 80°=5.67128 
200 



1134,25600=Iladius, 1134. 
Nat. CO. sine of 80° =,17365 



5671280 
6805536 
3402768 
7939792 
1134256 



196,96355440=Semi-chord, 197. 



Nat. sine of 80° 1134 

,9848 



9072 
4536 

9072 
10206 



lllfi 7fi.R9 Distance from centre to said 
J.1AU, (UO^ cliord=1117. 



104 The Surveyor's Guide. 

From 1134 
Take 1117 

17= Versed sine FE— From 197 
Take 10 

(187^=34969 
From (1134)^=1285956 
Take ( 187)^= 34969 



1250987=1118 
From which take 1117 

1 ft. the 1st ordinate. 

Again, for the 2d ordinate, 197— 20=(177)^ 

=31329 and 1285956 
— 31329 

(1120 From 1120 

v/ 1254627 Take 1117 

1 

— 3=the next. 

21)25 
21 

222)446 
444 



2240)227 

Ordinate. All this is plain from the figure, and 
"when the radius and constant subtrahend are 



The Surveyor's Guide. 



10^ 



found (which is on]j the work of a minute) all the 
others are nearly had at sight. This I consider 
quite superior to any other method now in practice. 
Otherwise thus : Let the radius, versed sine, 
chord, and constant quantity D E, be found as 







before, divide the semi-chord into any number of 
parts as e f g h i. From E C deduct one of the 
parts i C, leaves E i^m F, then D F=(radius) 
squared — (m F)^=(m Df the square root of which, 
minus the constant quantity, E D, gives the ordi- 
nate i F, and in like manner all the others are 
found, and thus the curve can be laid off in a few 
minutes in the most accurate manner, (by the 47th 
of the first of Euclid.) 



106 



The Sukveyor's Guide. 



Problem. 
Let A B C be a right angled triangle, the hy- 
pothenuse of which is 35, and the difference be- 
tween the area of the enscribed square (one of 
c whose angles coin- 
cides with the right 
angle of the trian- 
gle) and the area 
of the A is 150. 
Required the sides 
of the triangle. 




Solution. 

Put A E=x and D F=y. Then per 4th Euc. 
6th, y : y :: X : f =C F, f +x y=DoubIe the 
area of the Ag A D E and D F C .-.=300 or 'f + 
2 X y=600. Also f+ Y=B C and x+y= A 
B. Now (l'+jy+{x + yy=S5^ viz; |:+? + y^ 
+y+2 X y=1225 

Deduct 'f +2 X y=600 

5+2 y^+x^-=625 Ex't the square. 
Root and -^'+x=25 or 

y^+x^=25 x=A 1)^=5 ^/ x. 



The Surveyor's Guide. 107 

Hence 5 \/ x : x :: 35 : x+y, and by divi- 
ding the first and third by 5. x| : x :: y x+y 
and :: are their squares, x : x^ :: 49 . x^+y^+ 
2 X y. Multiply the extremes and means, and 

49 x2=x3+x yH-2 x^ y-j-by x and 
From 49 x =x''+y=2 xy ; but 25 x=j^+x^ .-. 
Take 25 x =x'+y^ 

24 X = 2 X y or 
24 = 2 y and 

y =12 the side of the square. 
And the sides of the angle are 21 and 28. 

28 
21 

28 
56 

2)588 

294=area of A. 
Deduct, 144 

150=area of the 2 A's. 



108 



The Surveyor's Guide. 



1 TABLES OP LATITUDE AND DEPARTURE. i 


1 
1 


N. S. 


E.W. 1|N. S. 


E.W. 


N. S. 


E.W. 


N. S. 


E.W. 

89i 




Oi 
0.9999 


19i 


Oi 
0.9999 


89i 


o.i 






0.0043 


0.0087 


0.9999 


0.0131 


2 






1.9999 


0.0087 


1.9999 


0.0174 


1.9998 


0.0262 


3 






2.9999 


0.0131 


2.9998 


0.0261 


2.9997 


0.0.392 


4 






3.9999 


0.0174 


3.9998 


0.0349 


3.9996 


0.0623 


5 






4.9999 


0.0218 


4.9998 


0.0436 


4.9995 


0.0654 


6 






5.9999 


0.0262 


5.9997 


0.0623 


6.9994 


0.0785 


7 






6.9999 


0.0305 


6.9997 


0.0611 


6.9993 


0.0916 


8 






7.9999 


0.0349 


7.9997 


0.0698 


7.9992 


0.1047 


9 

1 






8.9999 


0.0393 


8.9996 


0.0785 
88J 


8.9991 


0.1178 


1° 


89° 


li 

0.9997 


88S_ 
0.0218 


li 


li 


88i 
0.0305 


0.9998J0.0174 


0.9996 


0.0262 


0.9995 


2 


1.9997! 0.0349 


1.9995 


0.0436 


1.9993 


0.0523 


1.9990 


0.0610 


3 


2.9995|0.0523 


2.9993 


0.0654 


2.9989 


0.0785 


2.9986 


0.0916 


4 


3.9994 


0.0698 


3.9990 


0.0872 


3.9986 


0.1047 


3.9981 


0.1221 


6 


4.9992 


0.0872 


4.9988 


0.1090 


4.9982 


0.1309 


4.9976 


0.1527 


6 


5.9991 


0.1047 


5.9985 


0.1309 


5.9979 


0.1570 


6.9972 


0.1832 


7 


6.9989 


0.1221 


6.9983 


0.1627 


6.9976 


0.1832 


6.9967 


0.2137 


8 


7.9988 


0.1396 


7.9981 


0.1745 


7.9972 


0.2094 


7.9962 


0.2443 


9 

1 


8.9886 


0.1570 


8.9978 


0.1963 


8.9969 


0.2356 


8.9958 


0.2748 


2° 


88° 


2i 


87S 
0.0392 


2i 
0.9990 


87i 
0.0436 


2i 


87i 


0.9994,0.0349 


0.9992 


0.9988 


0.0479 




1.9987 0.0698 


1.9984 


0.0785 


1.9981 


0.0872 


1.9977 


0.0959 


3 


2.9981 0.1047 


2.9977 


0.1178 


2.9971 


0.1308 


2.9965 


0.1439 


4 


3.9975 0.1396 


3.9969 


0.1570 


3.9962 


0.1745 


3.9964 


0.1919 


.-i 


4.9969 0.1745 


4.9961 


0.1963 


4.9952 


0.2181 


4.9942 


0.2399 


6 


5.9963 


0.2094 


5.9954 


0.2355 


5.9943 


0.2617 


5.9931 


0.2878 


7 


6.9957 


0.2443 


6.9946 


0.2748 


6.9933 


0.3053 


6.9919 


0.3358 


a 


7.9951 


0.2792 


7.9938 


0.3141 


7.9924 


0.3489 


7.9908 


0.3838 


9 

1 


8.9945 


0.3141 

87° 


8.9930 
3i° 


0.3533 

86.1 
0.0567 


8.9914 


0.3926 


8.9896 


0.4318 


3° 


H 


86i 
0.0610 


3i 


86i 


0.9986 


0.0523 


0.9984 


0.9981 


0.9978 


0.0654 


? 


1.9973 


0.1047 


1.9968 


0.11.34 


1.9963 


0.1221 


1.9957 


0.1308 


3 


2.9959 


0.1570 


2.9952 


0.1701 


2.9944 


0.1831 


2.9936 


0.1962 


4 


3.9945 


0.2093 


3.9935 


0.2268 


3.9925 


0.22421 


3.9914 


0.2616 


fi 


4.9931 


0.2617 


4.9919 


0.2835 


4.9907 


0.3062 


4.9893 


0.3270 


fi 


5.9918 


0.3140 


5.9903 


0.3402 


5.9888 


0.3663 


5.9871 


0.3924 


7 


6.9904 


0.3664 


6.9888 


0.3968 


6.9869 


0.4273 


6.9850 


0.4578 


S 


7.9890 


0.4187 


7.9871 


0.4535 


7.9861 


0.4884 


7.9829 


0.6232 


9 


8.9877 
E.W. 


0.4710 

N. S. 


8.9866 
E.W. 


0.5102 


9.9832 
E.W. 


0.6494 

N. S. 


8.9807 
E.W. 


0.6886 


N. S. 


N. S. 



The Surveyor's Guide. 



109 



TABLES OP LATITUDE AND DEPARTURE. j 






N. S. 


E. W. 1 iN. S. 
86° 4i 


E.W. 


K S. 


E. W. 


N. S. 


E.W. 

85i 




4° 


85S 


4i 


85i 


41 




1 


0.9976 


0.0698' 0.9972 


0.0741 


0.9969 


0.0784 


0.9965 


0.0828 




2 


1.9951 


).1395l 11.9945 


0.1482 


1.9938 


0.1569 


11.9931 


0.1656 




3 


2.9927 0.2093ii2.9917' 


0.2223 


2.9907 


0.2354 


12.9897 


0.2484 




4 


3.9902 


0.2790! '3.9890 


0.2964 


3.9977 0.31381 


13.9863 


0.3312 




5 


4.9878 


0.34881 !4.9862 


0.3705 


4.9846 


0.3923 


4.9828 


0.4140 




6|5.9854 


0.4185 ,5.9835 


0.4446 


5.9815 


0.4707 


5.9794 


0.4968 




7l6.9829 


0.4883' 6.9807 


0.51871 


'6.9784 


0.5492 


6.9759 


0.5796 




8 


7.9805 0.5580 17.9780 


0.59281 


7.9753 


0.6277 


I7.9725 


0.6625 




1 9 


8.9780|0.6278l 


8.9752 
5i 


0.6670 

84|"- 


8.9722 
5i 


0.7061 


8.9691 


0.7453 






5° 


85° 1 


84i 


5i 


84i 




1 


0.9961 


0.0S71 


0.9958 


0.0915: 


0.9954 


0.0958 


0.9949 


0.1002 1 




2 


1.9923 


0.1743 


1.9916 


0.1830' 


|1.9908 


0.1917 


1.9899 


0.2004 




I ^ 


2.9884 


0.2615 


2.9874 


0.2745: 


2.9862 


0.2875 


2.9849 


0.3006 




4- 


3.9846 


0.3486 


3.9832 


0.3660: 


'3.9816 


0.38.34 


3.979^" 


0.4008 




1 5 


4.9808 


0.4358 


4.9790 


0.4575 


'4.9770 


0.4792 


4.974S 


0.5009 




6 


5.9769 


0.5229 


5.9 748 jo. 5490 


5.9724 


0.5751 


5.9698 


0.6011 






6.9731 


0.6101 


6.9706!o.6405 


6.9678 


0.6709 


6.9648 


0.7013 




' 8 


7.9692 


0.6972 


7.9664 


0.7320 


:7.9632 


0.7668 


7.9597 


0.8015 




; 9 

i 

1 


8.9654 


0.7844' 


8.9622 

i « 

0.9940 


0.8235 


:8.95S6 


0.8626 


8.9547 


0.9017 




6° 

0.9945 


84 


83i 


6i 


83i 
0.1132 


1 6i 


83i 
01175 




0.1045 


0.1088 


0.9935 


,0.9930 




1 2U.9S90 


0.2090 


1.9881 


0.2177 


1.9871 


0.2264 


1.9861 


0.2351 




3 2.9836 


0.3136 


,2.9821 


0.3266 


2.9807 


0.3396 


I2.9792 


0.3526 




i 4 


3.9781 


4181; 


3.9762 


0.4355 


3.9743 


0.4528 


'3.9723 


0.4701 




5 


4.9726 


0.5226 


14.9703 


0.5443 


4.9678 


0.5660 


14.9653 


0.5877 




fi 


5.9671 


0.6272 


15.9643 


0.6532 


5.9614 


0.6792 


15.9584 


0.7052 




7 


6.9616 


0.7317 


6.9584 


0.7621 


6.9550 


0.7924 


6.9515 


0.8228 




8 


7.9562 


0.8362 


'7.9524 


0.8709 


7.9486 


0.9056 


7.9445 


0.9403 




9 

1 


8.9507 


0.9408 
83° 


^8.9465 


0.9798 


8.9421 


1.0188 


8.9376 


1.0578 




7° 


n 


82i 


0.9914 


S2i 


1 ri 

0.9908 


82i 
0.1348 




0.9925 


0.1218 


0.9920 


0.1262 


0.1305 




2 


1.9851 


0.2437 


11.9840 


0.2524 


;i.9S29 


0.2610 


|1.9817 


0.2697 




3 


2.9776 


0.3656 


,2.976( 


0.3786 


:2.9743 


0.3916 


!2.9726 


0.4045 




4 


3.9702 


0.4874 


i.3.9680 


0.5048 


:3.9657 


0.5221 


13.9635 


0.5394 




5 


4.9627 


0.6093 


4.9600 


0.6310 


4.9572 


0.6526 


14.9543 


0.6742 




6 


5.9553 


0.7312 


5.952( 


0.7572 


5.9487 


0.7831 i 15.9452 


0.8091 




7 


6.9478 


0.8531 


16.9440 


0.8834 


6.9401 


0.9137 


6.9361 


0.9439 




8 


7.9404 


0.9750 


17.9360 


1.0096 


7.9315 


1.0442 


7.9269 


1.0788 




918.9329 


1.0968 


8.9280 


1.1358 


8.923(. 


1.1747 


8.9178 


1.2136 




|e.w. 


N. S. IE. W. 


N. S. 


E. W. 


N. S. 


E. W. 


N. S. 


1 







110 



The Surveyor's Guide. 



j TABLES OF LATITUDE AND DEPARTURE. jj 


1 


N. S. 


E.W. 


N. S. 


E. W. 
~81i~ 


N. S. 


E. W. 


N. S. 


E. W. 

"81i 
0.1521 


8° 
0.9902 


82 


8i 


8i 


81i 
0.1478 


8i 
0.9883 


0.1391 


0.9896 


0.1435 


0.9890 


2 


1.9805 


0.2783 


1.9793 


0.2870' 


1.9780 


0.2956 


1.9767 


0.3042 


3 


2.9708 


0.4175 


2.9689 


0.4305 


2.9670 


0.4434 


2.9651 


0.4564 


4 


3.9611 


0.5567 


3.9586 


0.5740 


3.9560 


0.6912 


3.9534 


0.6085 


5 


4.9513 


0.6959 


4.9483 


0.7175 


4.9451 


0.7390 


4.9418 


0.7606 


6 


5.9416 


0.8350 


5.9379 


0.8605 


5.9341 


0.8868 


5.9302 


0.9127 


7 


6.9319 


0.9742 


6.9276 


1.0045 


6.9231 


1.0347 


6.9186 


1.0649 


8 


7.9221 


1.1134 


7.9172 


1.1479, 


7.9121 


1.1825 


7.9069 


1.2170 


9 

1 


8.9124 


1.2526 


8.9069 


1.2914 


8.9011 


1.3303 


8.8952 

9i 
0.9855 


1.3691 

80i 
0.1693 


9° 


81 


9i 


80i 


9i 
0.9863 


804 


0.9877 


0.1564 


'0.9870 


0.1607 


0.1650 


2 


1.9754 


0.3129 


;L9740 


0.3215 


1.9726 


0.3301 


1.9711 


0.3387 


3 


2.963] 


0.4693 


:2.9610 


0.4822 


2.9589 


0.4951 


2.9566 


0.6080 


■4 


3.9508 


0.6257 


3.9480 


0.6430 


3.9451 


0.6602 


3.9422 


0.6774 


5 


4.9384 


0.7822 


4.9350 


0.8037 


4.9314 


0.8252 


4.9278 


0.8467 


6 


5.9261 


0.9386 


5.9220 


0.9644 


6.9177 


0.9903 


5.9133 


1.0161 


7 


6.9138 


1.0950 


6.9090 


1.1252 


,6.9040 


1.1553 


6.8989 


1.1854 


8 


7.9015 


1.2515 


7.8960 


1.2859 


17.8903 


1.3204 


7.8844 


1.3548 


9 

1 


8.8892 


1.4079 


,8.8830 


1.4467 


8.8766 


1.4864 

794 
0.1822 


8.8700 

101 
0.9824 


1.5241 

79i 
0.1866 


10° 


80° 
0.1736 


i lOi 


791 
0.1779 


! lOi 
11.9832 


0.9848 


0.9840 


2 


1.9696 


0.3473 


,L9681 


0.3559 


0.9665 


0.3645 


1.9649 


0.3730 


3 


2.9544 


0.5209 


2.9521 


0.5338 


2.9497 


0.5467 


2.9473 


0.6595 


4 


3.9392 


0.6946 


,3.9362 


0.7118 


13.9330 


0.7289 


3.9298 


0.7460 


s 


4.9240 


0.8682 


4.9202 


0.8897 


4.9163 


0.9112 


4.9123 


0.9325 


6 


5.9088 


1.0419 


5.9042 


1.0676 


5.8995 


1.0933 


5.8947 


1.1190 


7 


6.8937 


1.2155 


i 6.8883 


1.2456 


,6.8828 


1.2756 


6.8772 


1.3056 


8 


7.8785 


1.3892 


17.8723 


1.4235 


17.8660 


1.4579 


7.8596 


1.4920 


9 

1 


8.8633 


1.5628 


8.8564 


1.6015 

~7SF 
0.1951 


18.8493 

1 114 
0.9799 


1.6401 


8.8421 


1.6786 

m 

0.2036 


11° 


79 
0.1908 


1 Hi 


784 
0.1993 


111 


0.9816 


0.9808 


0.9790 


2 


L9633 


0.3816 


1.9616 


0.3902 


1.9598 


0.3987 


1.9581 


0.4073 


3 


2.9449 


0.5724 


'2.9424 


0.5853 


12.9398 


0.5981 


2.9371 


0.6109 


4 


3.9265 


0.7632 


13.9231 


0.7804 


13.9197 


0.7975 


3.9162 


0.8146 


5 


4.9081 


0.9540 


4.9039 


0.9755 


4.8996 


0.9968 


4.8952 


1.0182 


6 


5.8898 


1.1449 


5.8847 


1.1705 


5.8796 


1.1962 


6.8743 


1.2218 


y 


6.8714 


1.3357 


6.8655 


1.3656 


6.8595 


1.3956 


6.8533 


1.4265 


i 8 


7.8530 


1..5265 


7.8463 


1.5607 


7.8394 


1.5949 


7.8324 


1.6291 


1 9 

1 


8.8346 
E.W. 


1.7173 


8.8271 

eTw" 


1.7558 
N. S. 


8.8193 

ettv: 


1.7943 

N. S. 


8.8114 
E. W. 


1.8327 

nTsT 


N. S. 



The Surveyor's Guide. 



Ill 



TABLES OF LATITUDE AND DEPARTURE. 1 


1 


N. S. 

12° 

0.9781 


E.W. 


N.S. 
12i 


E. AY.I 

0.2122 


N. S. 


E. W. 


N. S. 


E. W. 

~77r 


78° 


12i 


77i 


12s 


0.2079 


0.9772 


0.9763 


0.2164 


0.9753 


0.2207 


2 


1.9563 


0.4158 


1.9544 


0.4242 


1.9526 


0.4329 


1.9507 


0.4414 


3 


2.9344 


0.6237 


2.9317 


0.6365 


2.9289 


0.6493 


2.9260 


0.6621 


4 


3.9126 


0.8316 


3.9089 


0.8487 


3.9052 


0.8657 


3.9014 


0.8828 


5 


4.S907 


L0396 


4.8861 


L0609 


4.8815 


1.0822 


4.8767 


L1035 


6 


5.8689 


1.2475 


5.8634 


1.2730 


5.8578 


1.2986 


5.8520 


1.3242 


7 


6.8470 


1.4554 


6.8406 


1.4852 


6.8341 


1.5151 


6.8274 


1.5449 


8 


7.8252 


1.6633 


7.8178 


1.6974 


7.8104 


1.7315 


7.8027 


1.7656 


9 

1 


8.8033 


1.8712 


8.7951 
13i 


1.9096 


8.7867 


1.9479 


8.7781 


1.9863 

76i 

0.2377 


13° 


77° 


76i 
0.2292 


0.9724 


76i 


13i 
0.9713 


0.9744 


O.2249I 


0.9734 


0.2334 


2 


1.9487 


0.4499 


L9467 


0.4584 


1.9447 


0.4669 


1.9427 


0.4754 


3 


2.9231 


0.6749 


2.9201 


0.6876 


2.9171 


0.7003 


2.9140 


0.7131 


4 


3.8975 


0.8998 


3.8934 


C.9168 


3.8895 


0.9338 


3.8854 


0.9507 


5 


4.8718 


1.1248 


4.8669 


1.1460 


4.8619 


1.1672 


4.8567 


L1884 


6 


5.8462 


1.34971 


5.8403 


1.3752 


5.8343 


1.4007 


5.8280 


1.4261 


7 


6.8206 


1.5746! 


6.8136 


1.6044 


6.8067 


1.6341 


6.7994 


1.6638 


8 


7.7950 


1.7996 


7.7870 


L8336 


7.7790 


1.8676 


7.7707 


1.9015 


9 
1 


8.7693 


2.0246 


8.7604 


2.0628 

75f 
0.2461 


8.7515 


2.1010 


8.7421 

141 
0.9670 


2.1392 

75i 
0.2546 


14° 


76° 


14i 
0.9692 


m 


75i 


0.9703 


0.2419 


0.9681 


0.2504 


2 


1.9406 


0.4838 


,1.9385 


0.4923 


L9363 


0.5008 


1.9341 


0.5092 


3 


2.9109 


0.7258 


2.9077 


0.7385 


2.9044 


0.7511 


2.9011 


0.7638 


4 


3.8812 


0.9677 


3.8769 


0.9846 


3.872711.0015 


3.8682 


1.1084 


5 


4.8515 


1.2096 


4.8461 


1.2308 


4.8407|l.2519 


4.8352 


1.2730 


fi 


.').8218 


1.4515 


5.815411.4769 


5.8089 1.5023 


5.8023 


1.5276 


7 


6.7921 


1.6935 


6.7846 


1.7231 


6.7770IL7527 


6.7693 


1.7822 


S 


7.7624 


1.9354 


7.7538 


1.9692 


7.7452 


2.0030 


7.7364 


2.0368 


9 

1 


8.7327 


2.1773 

75 
0.'25S8 


8.7231 


2.2154 
T4i" 


8.7133 


2.2534 


8.7034 


2.2914 


15 
0.9659 


15i 


15,} 


74i 
0.2672 


15i 
0.9624 


741 
0.2714 


0.9648 


0.2630 


0.9636 


?. 


1.9319 


0.5176 


1.9296 


0.5261 


1.9273 


0.5345 


1.9249 


0.5429 


3 


2.8978 


0.7765 


2.8944 


0.7891 


2.8909 


0.8017 


2.8874 


0.8143 


4 


3.8637 


1.0353 


3.8591 


1.0521 


3.8545 


1.0689 


3.8498 


1.0858 I 


ft 


4.8296 


1.2941 


4.8239 


1.3152 


4.8182 


L3362 


4.8123 


1.3572 1 


6 


5.7956 


1.5529 


5.7887 


1.5782 


5.7818 


1.6034 


5.7747 


1.6286 


7 


6.7615 


1.8117 


6.7535 


1.8412 


6.7454 


1.8707 


6.7372'l.9001 




7.7274 


2.0706 


7.7183 


2.1042 


7.7090 


2.1379 


7.6996 2.1715 


9 


8.6933 
E.W. 


2.3294 

N. S. 


8.6831 

bTw: 


2.3673 


8.6727 
E. W. 


2.4051 


8.6621 2.4430 


N.S.I 


N. S. 1 


E. W. 


N. S. !l 



112 



The Surveyor's Guide. 



TABLES OP LATITUDE AND DEPARTURE 




1 


N.S. 
16° 


E.W. 


N.S. 


E.W. 


N.S. 
16} 


E.W. 


N.S. 


E.W. 

73i 


74 
0.2756 


16i 


731 
0.2798 


73i 
0.2840 


16S 
0.9575 


0.9612 


0.9600 


0.9588 


0.2882 


2 


L9225 


0.5513 


1.9201 


).5596 


1.9176 


0.5680 


1.9151 


).5764 


3 


2.8838 


0.8269 


2.8801 


0.8395 


2.8765 


0.8520 


2.8727 


0.8646 


4 


3.8450 


L1025 


3.8402 


1.1193 


3.8353 


1.1361 


3.8303 


1.1528 


5 


4.8063 


L3782 


4.8002 


L3991 


4.7941 


1.4201 


4.7878 


1.4410 


6 


5.7676 


1.6538 


5.7603 


1.6790 


5.7529 


1.7041 


5.7454 


1.7292 


7 


6.7288 


1.9295 


6.7203 


1.9588 


6.7117 


1.9881 


6.7030 


2.0174 


8 


7.6901 


2.2051 


7.6804 


2.2386 


7.6705 


2.2721 


7.6606 


2.3056 


9 
1 


8.6513 


2.4807 


8.6404 


2.5186 


8.6294 


2.5661 

m 

0.3007 


8.6181 

m 

0.9523 


2.5938 


17 
0.9563 


73 


17i 


721 
0.2905 


17i 
0.9537 


72i 
0.3048 


0.2924 


0.9550 


?. 


1.9126 


0.5847 


1.9100 


0.5931 


1.9074 


0.6014 


1.9048 


0.6097 




2.8689 


0.8771 


2.8651 


0.8896 


2.8611 


0.9021 


2.8672 


0.9146 1 


4 


3.8252 


1.1695 


3.8201 


1.1862 


3.8149 


1.2028 


3.8096 


1.2195 1 


fi 


4.7815 


1.4619 


4.7751 


1.4827 


4.7686 


1.6035 


4.7620 


1.6243 i 


fi 


5.7378 


1.7642 


5.7301 


1.7792 


5.7223 


1.8042 


5.7144 


1.8292 1 


7 


6.6941 


2.0466 


6.6851 


2.0758 


6.6760 


2.1049 


6.6668 


2.1340 


8 


7.6504 


2.3390 


7.6402 


2.3723 


7.6297 


2.4056 


7.6192 


2.4389 


9 
1 


8.6067 


2.6313 


8.5952 

18i 
0.9497 


2.6689 


8.6834 


2.7063 


8.6716 


2.7438 


18° 


72 
0.3090 


711 
0.3131 


18J 
0.9483 


71i 


18i 
0.9469 


7U 


0.9510 


0.3173 


0.3214 1 


? 


L9021 


0.6180 


1.8994 


0.6263 


1.8966 


0.6346 


1.8939,0.6429 || 


3 


2.8532 


0.9271 


2.8491 


0.9395 


2.8460 


0.9619 


2.8408 


0.96-13 


4 


3.8042 


1.2361 


3.7988 


1.2527 


3.7933 


1.2692 


3.7877 


1.2857 


ft 


4.7553 


1.5451 


4.7485 


L6658 


4.7416 


1.5865 


4.7346 


1.6072 


fi 


5.7063 


1.8541 


5.6982 


1.8790 


5.6899 


L9038 


6.6816 


1.9286 


7 


6.6574 


2.1631 


6.6479 


2.1921 


6.6383 


2.2211 


6.6285 


2.2501 


8 


7.6084 


2.4721 


7.6976 


2.5053 


7.6866 


2.5384 


7.6764 


2.6716 


9 
1 


8.5595 


2.7812 

71° 
0.3255 


8.5473 


2.8185 

7oa 

0.3297 


8.6349 


2.8567 


8.6224 


2.8929 
.70i 
0.3379 


19° 
0.9455 


19i 


70i 
073338 


19i 
0.9412 


0.9441 


0.9426 


9 


1.8910 


0.6511 


1.8882 


0.6594 


1.8853 


0.6676 


1.8823 


0.6768 


3 


2.8366 


0.9767 


2.8323 


0.9891 


2.8279 


1.0014 


2.8233 


1.0137 


4 


3.7821 


1.3023 


3.7764 


1.3188 


3.7706 


1.3362 


3.7647 


1.3517 


^ 


4.7276 


1.6278 


4.7204 


1.6484 


4.7132 


1.6690 


4.7069 


1.6896 


5.6731 


1.9534 


5.6645 


1.9781 


5.6568 


2.0028 


5.6471 


2.0275 


7 


6.6186 


2.2790 


6.6086 


2.3078 


6.6985 


2.3366 


6.5882 


2.3654 


8 


7.5641 


2.6045 


7.5527 


2.6375 


7.6411 


2.6705 


7.5294 


2.7033 


9 


8.5097 

eTw: 


2.9301 
N.S. 


8.4968 
E.W. 


2.9672 

N.S. 


8.4838 


3.0043 

N.S. 


8.4706 
E.W. 


3.0412 


E.W. 


N.S. 



The Surveyor's Guide. 



113 



tables oe latitude and departure. I 




1 


N. S. 
20 


E.W. 

70 


N. S. 1 
"201" 
0.9382 


E.W. 


N. S. 


e. w. 

69 i 
0.3502 


N. S. 


E.W. 

69i 
0.3543 




69S 


20i 


201 




0.9397 


0.3420 


0.3461 


0.9366 


0.9351 




t 


1.8794 


0.6840 


1.8764 


0.6922 


1.8733 


0.7004 


1.8703 


0.7086 




•6 


2.8191 1.0261] 


2.8146 


1.0383 


2.8100 


1.0606 


2.8054 


1.0629 




4 


3.7588 1.36811 


3.7528 


1.3845 


3.7467 


1.4008 


3.7405 


1.4172 




6 


4.6985 


1.7101 


4.6910 


1.7306 


4.6834 


1.7510 


4.6757 


1.7715 




6 


5.6381 


2.0521 


5.6291 


2.0767 


6.6200 


2.1012 


5.6108 


2.1267 




7 


6.5778 


2.3941 


6.5673 


2.4228 


6.6567 


2.4614 


6.5459 


2.4800 




8 


7.5175 


2.7362 


7.5055 


2.7689 


7.4934 


2.8016 


7.4811 


2.8343 




9 
1 


8.4572 


3.0782 


8.4437 


3.1160 


8.4300 


3.1519 


8.4162 


3.1886 
68i 




21° 


69° 


21i 


68i 
0.3624 


2U 


68i 


211 




0.9336 


0.3583 


0.9320 


0.9304 


0.3665 


0.9288 


0.3705 




2 


1.8672 


0.7167 


1.8640 


0.7249 


1.8608 


0.7330 


1.857610.7411 1 




3 


2.8008 


L0751 


2.7960 


1.0873 


2.7913 


1.0995 


2.7864 1.1117 




4 


3.7343 


1.4335 


3.7280 


1.4497 


3.7217 


1.4660 


3.7162 1.4822 




5 


4.6679 


L7918 


4.6600 


1.8122 


4.6521 


1.8325 


4.6440 1.8528 




6 


5.6015 


2.1502 


15.5920 


2.1746 


5.5825 


2.1990 


6.6729 2.2233 




7 


6.5351 


2.5086 


6.5240 


2.5371 


6.5129 


2.6665 


6.5017 2.5939 




8 


7.4686 


2.8669 


7.4560 


2.8995 


7.4433 


2.9320 


7.4305 2.9644 




9 

1 


8.4022 


3.2253 


8.3880 


3.2619 


8.3738 


3.2985 

67i 

0.3827 


8.3593 


3.3360 




22° 


68° 


22i 


67i 


22J 


22i 


67i 




0.9272 


0.3746 


0.9255 


0.3786 


0.9239 


0.9222 


0.3867 




2 


1.8544 


0.7492 


1.8511 


0.7673 


1.8478 


0.7654 


1.844410.7734 




3 


2.7816 


L1238 


2.7766 


1.1359 


2.7716 


1.1480 


2.7666 


1.1601 




' 4 


3.70S7 


1.4984 


3.7022 


1.5146 


3.6956 


L5307 


3.6888 


1.5468 




fi 


4.6359 


1.8730 


4.6277 


1.8932 


4.6194 


1.9134 


4.6110 


1.9335 




6 


5.5631 


2.2476 


5.5532 


2.2719 


6.5433 


2.2961 


5.6332 


2.3202 


1 


7 


6.4903 


2.6222 


6.4788 


2.6505 


6.4671 


2.6788 


6.4564 


2.7069 




i 8 


7.4175 


2.9968 


7.4043 


3.0292 


7.3910 


3.0615 


7.3776 


3.0936 




1 ' 

i 1 


8.3447 
23 


3.3715 


8.3299 


3.4078 


8.3149 


3.4441 


8.2998 


3.4803 

66i 

(L4027 




67 


23i 


66S 


23i 


66i 
0.3987 


23i 




0.9205 


0.3907 


0.9188 


0.3947 


0.9170 


10.9163 




. 5> 


1.8410 


0.7815 


1.8376 


0.7895 


1.8341 


0.7976 


|l.8306 


0.8055 






2.7615 


1.1722 


2.7564 


1.1842 


2.7612 


L1962 


2.7469 


1.2082 




4 


3.6820 


1.5629 


3.6762 


1.5790 


3.6682 


L6960 


3.6612 


1.6110 




5 


4.6025 1.9537 


4.5939 


1.9737 


4.5853 


11.9937 


4.5766 


0.0137 




fi 


5.5230 2.3444 


5.5127 


2.3685 


6.6024 


2.3925 


5.4919 


2.4165*! 1 


7 


6.4435 2.7351 


6.4315 


2.7632 


6.4194 


2.7912 


6.4072 


2.8192 \ 


S 


7.3640 3.1258 


7.3603 


3.1579 


7.3365 


3.1900 


7.2226 


3.2220 




9 

i 


8.2845|3.5166 


8.2691 

'e.'w 


3.6527 


8.2636 


3.5887 
N. S. 


8.2378 


3.6247 
N. S. 




e. W 


IN.S. 


N. S. 


E.W 


E.W 


J 



114 



The Surveyor's Guide. 



TABLES OF LATITUDE AND DEPARTURE. jl 


1 

1 


N. S. 


E.W. 


N. S. 


E. W. 
651 
0.4107 


N. S. E. W.| 


N. S. 


E. W. 
'"651" 
0.4186 


24 
0.9135 


66 


24i 


24i 


65i 


24i 


0.4067 


0.9117 


0.9099 0.4147 


0.9081 


?. 


1.8271 


0.8135 


1.8235 


0.8214 


1.8199 0.8294 


1.8163 


0.8373 


8 


2.7406 


1.2202 


2.7353 


1.2322 


2.7299 1.2441 


2.7244 


L2560 


4 


3.6542 


1.6269 


3.6470 


1.6429 


3.6398 1.6588| 


3.6326 


1.6746 


ft 


4.5677 


2.0337 


4.5588 


2.0536 


4.5498 


2.0735 


4.5407 


2.0933 1 


6 


5.4813 


2.4404 


5.4706 


2.4643 


5.4598 


2.4882 


5.4489 


2.5122 


7 


6.3948 


2.8472 


6.3823 


2.8750 


6.3697 


2.9029 


6.3570 


2.9306 


1 ^ 


7.3084 


3.2539 


7.2941 


3.2857' 


7.2797 


3.3175 


7.2651 


3.3493 


9 
1 


8.2219 


3.6606 


8.2058 


3.6965 


8.1896 


3.7322 


8.1733 


3.7679 

64i 1 
0.4344 


25° 


65 


25i 


64i 


25i 


64i 


25i 
0.9007 


0.9063 


0.4226 


0.9044 


0.4265 


0.9026 


0.4305 




1 8126 


0.8452 


|L8089 


0.85311 


1.8052 


0.8610 


1.8014 


0.8688 


3 


2.7189 


1.2679 


2.7134 


1.2797 


2.7077 


1.2915 


2.7021 


1.3032 


4 


3.6252 


1.6905 


3.6178 


L7063 


3.6103 


1.7220 


3.6028 


1.7376 


ft 


4.5315 


2.1131 


4.5223 


2.1328 


4.5129 


2.1525 


4.5035 


2.1720 


fi 


5.4378 


2.5357 


15.4267 


2.5594 


5.4155 


2.5831 


15.4042 


2.6064 


7 


6.3442 


2.9583 


6.3312 


2.9860 


16.3181 


3.0136 


6.3049 


3.0408 


S 


7.2505 


3.3809 


7.2356 


3.4125 


17.2207 


3.4441 


7.2056 


3.4752 


9 
1 


8.1568 


.3.8036 


8.1401 


3.839] 


8.1233 
0.8949 


3.8746 


8.1063 


3.9096 

63i 
0.4501 


26° 


64° 


26i 


634 
0.4423 


63i 


26.! 
0.8930 


0.8988 


0.4384 


0.8969 


0.4462 




1.7976 


0.8767 


[1.7937 


0.8846 


!1.7899 


0.8924 


1.7859 


0.9002 


1 S 


2.6964 


1.3151 


2.6906 


1.3269 


|2.6848 


1.3386 


2.6789 


1.3503 


4 


3.5952 


1.7535 


13.5875 


L7692 


13.5797 


1.7848 


3.5719 


1.8004 


ft 


4.4940 


2.1919 


14.4843 


2.2115 


4.4746 


2.2310 


4.4649 


2.2505 


6 


5.3928 


2.6302 


15.3812 


2.6537 


5.3696 


2 6772 


5.3579 


2.7006 


7 


6.2916 


3.0686 


.6.2781 


3.0960 


6.2645 


3.1234 


6.2508 


3.1507 


8 


7.1904 


3.5070 i7.1750 


3.5383 


7.1594 


3.5606 


7.1438 


3.6008 


9 
1 


8.0891 


3.9453 

_6_3_ 

0.4540 


18.0718 

I'm' 

' 0.8890 


3.9806 
0.4578 


8.0544 


4.0158 


8.0368 


4.0509 


27i 


62i- 
0.4617 


271 
0.8850 


62i 
0.4056 


0.8910 


0.8870 


? 


1 7820 


0.9080 


'1.7780 


0.9157 


1.7740 


0.9235 


1.7700 


0.9312 


3 


2.0730 


1.3620 


2.6670 


1.3736 


,2.6610 


,L3852 


2.6550 1.3968 \ 


^ 4 


3.5640 


L8160 


3.5561 


1.8315 


3.5480 


L8470 


3.5400 


1.8624 1 


1 '"^ 


4.4550 2.2699 


4.4451 


2.2894 


4.4350 


2.3087 


4.4250 


2.3281 


1 fi 


5.3460 


2.7239 


5.3341 


2.7472 


5.3221 


2.7705 


5.3099 


2.7937 


1 7 


6.2370 


3.1779 


6.2231 


3.2051 


6.2092 


3.2322 


6.1949 


3.2593 


8 


7.1280 


.3.6319 


7.1121 


3.6030 


7.0961 


3.6940 


7.0799 


3.7249 


9 


8.0191 
E.W. 


4.0859 
N. S. 


8.0011 
1 E.W. 


4.1209 
N. S. 


7.9S31 


4.1553 

N. S.- 


7.9049 

!e. W. 


4.1905 

'n. S. 


E.W. 



The Surveyor's Gqide. 



115 





TABLES OP LATITUDE 


AND 


DEPARTURE 




1 


N. S. 

28 
0.8829 


E. W. 

62 
0.4694 


N. S. 
28i 


E. AV.i 

611 
0.4733 


N. S. 


E.W. 


N. S. 

2SJ 

0.8766 


E.W. 


28,^ 


6U 


61i 


0.8809 


0.8788 


0.4771 


0.4810 


2 


L7659 


0.9389 


L7618 


0.9466 


1.7576 


0.9543 


L7534 


0.9620 


3 


2.64S8 


1.4084 


2.6427 


1.4199 


2.6364 


1.4315 


2.6302 


1.4430 


4 


3.5318 L8779! 


3.5236 


L8933 


3.5153 


1.90S6 


3.5069 


1.9239 


5 


4.4147 


2.3474 


4.4045 


3.3666 


4.3941 


2.3858 


4.3836 


2.4049 


fi 


5.2977 


2.8168 


5.2854 


2.8399 


5.2729 


2.8629 


5.2604 


2.8859 


7 


6.1806 


3.2863 


6.1662 


3.3132 


6.1517 


3.3401 


6.1371|.3.3669 


8 


7.0636 


3.7558 


7.0471 3.78661 


7.0305 


3.8173 


7.013S 3.8479 


9 

1 


7.9465 


4.2252 


7.9280 4.2599 
29J 1 60i 


7.9093 


4.2944 
60i 


7.8905 

29i 

0.8682 


4.3289 


29 


61 
0.4848 


29i 
0.8703 


60i 


0.8746 


:0.S725 


0.4SS6 


0.4924 


0.4962 


2 


L7492 0.9696] 


,L7450 


0.9772 


1.7407 


0.9848 


1.7364 


0.9924 


3 


2 6239 


1.45441 


,2.6175 


1.4659 


2.6111 


1.4773 


2.6046 


1.4886 


4 


3.4985 


1.9392 


3.4900 


1.9545 


3.4814 


1.9697 


3.4728 


1.9849 


5 


4.3731 


2.4240 


4.3625 


2.443 ll 


4.3518 


2.4621 


4.3410 


2.481] 


fi 


5.2477 


2.9089 


'5.2350 


2.93171 


5.2221 


2.9545 


5.2092 


2.9773 


7 


6.1223 


3.3937 


(6.1075 


3.4203 


6.0925 


3.4463 


6.0774 


3.4735 


S 


6.997013.8785 


6.9S00 


3.9090i 


6.9628 


3.9394 


7.9456 


3.9697 


9 

1 


7.8716J4.3633 


7.8525 

30i 
0.8638 


4.3976 
, 59i 
i0.5038 


7.8332 

30i- 

0.8616 


4.4318 


7.8138 
30i 


4.4669 


30 


60 
0.5000 


59i 
0.5075 


59i 


0.S660 


0.8594 


0.5113 


*? 


I.7320I1.OOOO 


L7277 


!l.0076 


1.7232 


1.0151 


L7188IL0226 


8 


2.59S1 


1.5000 


2.5915 


1.5113 


2.5849 


1.5226 


2.57S2;L5339 


4 


3.4041 


2.2000 


3.4552 


2.0151 


3.4465 


2.0301 


3.4376 


2.0452 


,T 


4.3301 


2.5000 


4.3192 


2.5189 


4.3081 


2.5377 


4.2970 


2.5564 


fi 


5.1961 


3.3000 


5.1830 


3.0226 


5.1698 


3.0452 


5.1564 


3.0677 


7 


6.0622 


3.5000 


6.0468 


!3.5264 


16.0314 


3.5528 


6.0158 


3.5790 




6.92S2 


4.0000 


6.9107 


4.0302 


6.8930 


4.0603 


6.8752 


14.0903 


9 
1 


7.7942 


4.5000 

59 
0.5150 


7.7745 

31i 

0.8549 


4.5339 
0.5188 


:7.7547 


4.5678 


7.7346 

31i 

0.8503 


4.6016 


1-^ 
0.8571 


31i 


68i 
0.5225 


58i 


0.8526 


0.5262 


'>. 


1.7143 


L0301 


1.709S 


1.0375 


L7053 


L0450 


1.7007 


1.0524 


q 


2.5715 


1.5451 


2.5647 


1.5563 


2.5579 


1.5675 


2.5510iL5786 


/| 


13.4287 


2.0602 


3.4196 2.0751 


3.4106 


2.0900 


3.4014 2.1048 


5 


14.2858 


2.5752 


4.2745 2.5939 


4.2632 


2.6125 


4.251S 


2.6311 


fi 


5.1430 


3.0902 


5.1295:3.1126 


5.1158 


3.1350 


5.1021 


3.1573 




16.0002 


3.6053 


5.98443.6314 


5.9685 


3.6575 


5.9525 


3.6835 


8 


6.8573 


4.1203 


16.8393 4.1502 


6.8211 


4.1800 


6.8028 


4.2097 


9 


7.7145 


4.6353 


7.694214.0689 


7.6738 


4.7025 


7.6532 


4.7359 




E. W 


N. S. 


E.W 


1 N. S. 


E.W. 


1 N.S. 


E.W 


1 N.S. 



116 



The Surveyors Guide. 





TABLES OF LATITUDE AND 


DEPARTURE 






1 


N. S. 


E.W. 
58° 


N. S. 


E.\Y. 


N. S. 


E.W. 

57i 

0.5373 


N. S. 

32i 

0.8410 


E.W. 
"67i" 
0.6409 




32° 
0.8480 


32i 
0.8457 


57S 


32i 
0.8434 




0.5299 


0.6336 




2 


1.6961 


1.0598 


L6914 


1.0672 


1.6868 


1.0746 


L6821 


1.0819 




•6 


2.5441 


L5897 


2.5372 


1.6008 


2.6302 


1.6119 


2.5231 


L6229 




4 


3.3922 


2.1197 


3.3829 2.1344 


3.3736 


2.1492 


3.3642 


2.1639 




5 


4.2402 


2.6496 


4.2286 2.6681 


4.2169 


2.6865 


4.2052 


2.7049. 




6 


5.0883 


3.1795 


5.0744 


3.2017 


5.0603 


3.2238 


5.0462 


3.2458' 




1 


5.9363 


3.7094 


5.9201 


3.7353 


5.9037 


3.7611 i 


5.8873 


3.7868 




8- 


6.7844 


4.2394 


6.7658 


4.2689 


6.7471 


4.29841 


6.7283 


4.3278 




9 
1 


7.6324 


4.7693 


7.6115 


4.8026 
56S 


7.5905 


4.8357 


7.5694 
33i 


4.8688 
'"56i 
0.5566 




33 


57 


33i 


33i 
0^8339 


56^ 




0.8386 


0.6446 


0.8363 


0.6483 


0.6519 


0.8314 




2 


1.6773 


L0893 


1.6726 


L0966 


L6678 


1.1039 


1.6629 


Lllll 




3 


2.5160 


L6339 


2.6089 


1.6449 


2.5017 


1.6668 


2.4944 


L6667 




4 


3.3547 


2.1786 


3.3451 


2.1932 


3.3365 


2.2077 


3.3259 


2.2223 




5 


4.1934 


2.7232 


4.1S14 


2.7415 


4.1694 


2.7697 


4.1573 


2.7778 




6 


5.0320 


3.2678 


5.0177 


3.2898 


5.0033 


3.3116 


4.9888 


3.3334 




7 


5.8707 


3.8125 


5.8540 


3.8381 


5.8372 


3.8636 


5.8203 


3.8890 




8 


6.7094 


4.3571 


6.6903 


4.3863 


6.6711 


4.4165 


6.6618 


4.4446 




9 
1 


7.5480 


4.9018 
56° 


7.5266 


4.9346 


7.5069 


4.9674 


7.4832 


6.0001 




34 


34i 


65S 


34i 


55i 
0.5664 


34S 


56i 




0.8290 


0.5592 


0.8266 


0.5628 


0.8241 


0.8216 


0.5700 




2 


L65S1 


L1184 


L6632 


1.1256 


1.6482 


1.1328 


1.6433 


1.1400 




1 3 


2.4871 


1.6776 


2.4798 


1.6884 


2.4724 


L6992 


2.4649 


1.7100 




4 


3.3162 


2.2368 


3.3063 


2.2512 


3.2965 


2.2666 


3.2866 


2.2800 




5 


4.1452 


2.7960 


4.1329 


2.8140 


4.1206 


2.8320 


4.1082 


12.8500 




6 


4.0742 


3.3552 


4.9595 


3.3768 


4.9447 


3.3984 


4.9299 


13.4200 




7 15.8033 


3.9144 


5.7861 


3.9396 


5.7689 


3.9648 


5.7515 


3.9900 




8 1 6.6323 


4.4735 


6.6127 


4.5024 


6.5930 


4.5313 


6.5732 


4.5600 




9 

1 


7.4613 
■350 


5.0327 
55 


7.4393 


5.0662 


7.4171 
35i 


5.0977 

54i 

0.5807 


7.3948 


5.1300 
"Hi" 
0.6842 




35i 


541 


351 




0.8191 


0.6736 


0.8166 


0.5771 


0.8141 


0.8116 




1^ 


1.6383 


1.1472 


1.6333 


1.1543 


1.6282 


LI 614 


1.6231 


L1685 




.S 


2.4575 


1.7207 


2.4499 


1.7314 


2.4423 


L7421 


2.4347 


1.7527 




4 


3.2766 


2.2943 


3.2666 


2.3086 


3.2565 


2.3228 


3.2463 


2.3370 




f> 


4.0968 


2.8679 


4.0832 


2.8867 


4.0706 


2.9035 


4.0679 


2.9212 




fi 


4.9149 


3.4415 


4.899? 


3.4629 


4.8847 


3.4842 


4.8694 


3.5055 




7 


5.734] 


4.0160 


5.7165 


4.0400 


5.6988 


4.0649 


5.6810 


4.0897 




a 


6.5632 


4.5886 


6.5331 


4.6172 


6.5129 


4.6466 


6.4926 


4.6740 




9 


7.3724 


6.1622 

n". S. 


7.349S 


6.1943 


7.3270 

fETw: 


5.2263 

N. S. 


7.3042 


5.2582 

NTs. 




E. W. 


N. S. 


E.W 


. 



The Surveyor's Guide. 



117 



TABLES OF LATITUDE AND DEPARTURE. j 


1 

! 1 


N. S. 


E.W. 


N. S. 

36i 

0.8064 


E. AV. 

"63i~ 
0.5913 


N. S. 


E. W. 

53i" 


N. S. 


E. W. 

•53i 
0.5983 


36 

0.8090 


64 


364 


36S 
0.8012 


0.6878 


0.803810.5948 


\ 2 


1.6181 


1,1766 


1.6129 


1.1826 


1.6077|l.lS96 


1.6025 


1.1966 


3 


2.4271 


1.7634 


2.4193 


1.7739 


2.4116 


1.7845 


2.4038 


1.7950 


4 


3.2361 


2.3511 


3.2268 


2.3662 


3.2154 


2.3793 


3.2050 


2.3933 


5 


4.0451 


2.9389 


4.0322 


2.9566 


4.0193 


2.9741 


4.0063 


2.9916 


6 


4.8641 


3.6267 


4.8387 


3.6478 


4.8231 1.3.6689 


4.8075 


3.5899 


7 


6.6631 


4.1145 


6.6461 


4.1391 


6.6270 


4.1638 


5.6088 


4.1883 


8 


6.4721 


4.7023 


6.4516 


4.7304 


0.4308 


4.7686 


6.4100 


4.7866 


9 
1 


7.2812 


6.2901 
63 


7.2580 


6.3217 

521 
0.6053 


7.2347 


5.3534 


7.2111 


5.3849 

52i 
0.6122 


37 
0.7986 


37J 


37i 
0.7933 


62i 
0.6087 


371 
0.7907 


0.6018 


0.7960 


2 


1.6973 


1.2036 


1.5920 


1.2106 


1.5867 


1.2176 


1.5814 


1.2244 


3 


2.3959 


1.8054 


2.3880 


1.8159 


2.3801 


1.8263| 


2.3721 


1.8366 


4 


3.1945 


2.4073 


3.1840 


2.4212 


3.17.34 


2.4350 


3.1628 


2.4489 


5 


3.9932 


3.0091 


3.9800 


3.0266 


3.9668 


3.0438 


3.9534 


3.0611 


6 


4.7918 


.3.61091 


14.7760 


3.6318 


4.7601 


3.6526 


4.7441 


3.6733 


7 


5.6904 


4.2127 


15.5720 


4.2371 


5.6635 


4.2613 


5.6348 


4.2855 


8 


6.3891 


4.8145 


6.3680 


4.8424 


6.3468 


4.8701 


6.3265 


4.8977 


9 
1 


7.1877 


6.4163 


7.1640 


6.4476 


7.1402 


5.4788 


7.1162 


5.5099 


38° 


62 


38i 


611 1 
0.6191 


51i 
0.6225 


38i 


61i' 
0.6269 


0.7880 


0.6156 


0.7853 


0.7826 


0.7799 


2 


1.5760 


1.2313 


1.5706 


1.2382 


1.5662 


1.2450 


1.5598 


1.2518 


3 


2.3640 


1.8470 


2.3559 


1.8673 


2.3478 


1.8676 


2.3397 


1.8778 


4 


.3.1520 


2.4626 


3.1413 


2.4764 


3.1304 


2.4900 


3.1195 


2.5037 


5 


3.9401 


3.0783 


3.9266 


3.0955 


3.9130 


3.1125 


3.8994 


3.1296 


6 


4.7281 


3.6940 


4.7119 


3.7146 


4.6956 3.736] 


4.6793 


3.7555 


7 


5.5161 


4.3096 


5.4972 


4.3337 


5.4782|4.3576 


5.4692 


4.3815 


8 


6.3041 


4.9253 


6.2826 


4.9528 


6.2608 


4.9801 


6.2391 


6.0074 


9 
1 


7.0921 


6.5409 


7.0678 


5.5718 

ToF 


7.0434 


6.6026 


7.0190 


6.6333 

50i 
0.6394 


39° 


61 

0.62931 


39i 
0.7716 


50i 
0.6361 


39i 
0.7688 


0.7771 


0.7744 


0.6327 


9. 


1.5543 


1.2586 


1.5488 


1.2654 


1.5432 


1.2621 


1.6377 


1.2789 


3 


2.3314 


1.8880 


2.3232 


1.8981 


2.3149 


1.9082 


2.3065 


1.9183 


4 


3.1086 


2.5173 


3.0976 


2.6308 


3.0865 2.64431 


3.0764 


2.6678 1 


1 •"> 


3.885713.1466 


3.8719!3.1635| 


3.858ll3.1804| 


3.8442 3.1972 i 


1 fi 


4.6629 


3.7759 


4.6463 


3.7962 


4.6297 


3.8165 


4.613013.8366 j 


1 ^ 


5.4400 


4.4052 


5.4207 


4.4289 


5.4014 


4.4525] 


6.3819'4.4761 | 


8 


6.2172 


5.0346 


6.1951 


6.0616 


6.1730 


6.08861 


6.1507 


5.1165 


9 


6.9943 


5.6639 


6.9695 


5.6943 


6.9446 


5.7247 


6.9196 
E. W. 


5.7650 


E.W. 


N. S. 


E.W. 


N. S. 


E.W. 


N. S. 


N. S. 



118 



The Surveyor's Guide. 



TABLES OF LATITUDE AND DEPARTURE. [ 


1 


N. S. 

40° 

0.7660 


E. W. 
50 

0.6428 


N. S. 


E.W. 


N. S. 


E.W. 


N.S. 


E.W. 


40i 
0.-7632 


49i 


m 


49i 


401 
0.7676 


49i 


0.6461 


0.7604 


0.6494 


0.6527 


2 


1.5321 


1.2856 


L5265 


1.2922 


1.5208 


1.2989 


1.5161 


1.3055 


'6 


2.2981 


1.9284 


2.2897 


L9384 


2.2812 


1.9483 


2.2727 


1.9583 


4 


3.0642 


2.5711 


3.0529 


2.5845 


3.0416 


2.5978 


3.0303 


2.6110 


1 ^ 


3.8302 


3.2139 


3.8162 


3.2306 


3.8020 


3.2472 


3.787-8 


3.2638 


6 


4.5963 


3.8567 


4.5794 


3.8767 


J4.5624 


3.8967 


4.6464 


3.9166 


7 


5.3623 


4.4995 


5.3425 


4.6229 


6.3228 


4.6461 


6.3029 


4.6693 


8 


6.1284 


5.1423 


6.1059 


5.1690 


|6.0832 


5.1956 


6.0605 


5.2221 


9 
1 


6.8944 


5.7851 


6.8691 


5.8151 

481 


6.8436 


6.8460 
48i 


6.8181 


5.8748 
48i 
6.6659 


41 


49 


41i 
0.7518 


m 


411 
0.7460 


0.7547 


0.6560 


0.6593 


0.7489 


0.6626 


2 


1.5094 


1.3121 


L5037 


1.3187 


1.4979 


1.3252 


L4921 


1.3318 


3 


2.2041 


1.9682 


2.2555 


1.9780 


2.2468 


L9879 


2.2382 


1.9976 


4 


3.0188 


2.6242 


3.0074 


2.6374 


2.9958 


2.6505 


2.9842 


2.6635 


5 


3.7735 


3.2803 


3.7592 


3.2967 


3.7447 


3.3131 


3.7303 


3.3294 


6 


4.5283 


3.9364 


4.5110 


3.9660 


4.4937 


3.9757 


4.4764 


3.9953 


7 


5.2830 


4.5924 


6.2629 


4.6154 


6.2426 


4.6383 


5.2224 


4.6612 


8 


6.0377 


5.2485 


[6.0147 


6.2747 


5.9916 


6.3010 


5.9686 


6.3270 


9 
1 


6.7924 


5.9045 


6.7666 


5.9341 


6.7406 


5.9636 


6.7145 
'42i 


6.9929 


42 


48 
0.6691 


42i 
0.7402 


471 
0.6723 


,0.7373 


m 

0.676-6 


47i 


0.7431 


0.7343 


0.6788 


2 


1.4863 


1.3383 


1.4804 


1.3447 


11.4740 


1.3512 


1.4686 


1.3576 


3 


2.2294 


2.0074 


'2.2207 


2.0171 


2.2118 


2.0268 


2.2029 


2.0364 


4 


2.9726 


2.6765 


2.9609 


2.6895 


2.9491 


2.7024 


2.9373 


2.7152 


5 


3.7157 


3.34571 


3.7011 


3.3618 


3.6864 


3.3779 


3.6716 


3.3940 


6 


4.4589 


4.0148 


4.4413 


4.0342 


4.4237 


4.0535 


4.4059 


4.0728 


7 


5.2020 


4.6839 


5.1815 


4.7066 6.1610 


4.7291 


5.1402 


4.7516 


8 15.9452 


5.3530 


5.9218 


5.3789, 5.8982 


6.4047 


6.874616.4304 || 


9 


6.6883 


6.0222 
47 


6.6620 
0.7283 


6.0513 

46i 
0.6852 


6.6355 
1 43i 


6.0803 


6.6089 


6.1092 


1 


43 
0.7313 


m\ 


431 
0.7223 


46i 
0.69i5 


0.6S2oi 


0.7253'0.6883| 


;^ 


1.4627 


1.3640| 


1.45G7 


1.3704 


1.4507 


1.3767 


1.4447 


1..3830 


3 


2.1941 


2.0460 2.1851 


2.05561 ;2.1761 


2.0651 


2.1671 2.0745 !j 


4 


2.9254 


2.7280 2.9135'2.7407; '2.9015 


2.7534 


2.8894 2.7660 1 


5 3.6568 


3.4100 3.6418 3.4259| 


3.6269 


3.4418 


3.6118 3.4576 || 


6 i4.3881 


4.09201 4.3702 


4.1111 


4.3622 


4.1301 


4.3342 


4.1491 


7 5.1195 


4.7740 5.0986 


4.7963 


5.0776 


4.8186 


6.0565 


4.8406 i 


8 5.8508 


5.4560 5.8269 


6.4814 


6.8030 


6.5068 


6.7789 


5.5321 1 


9 6.5822 


6.1380! 6.5553 


6.1666 

N. S. 


6.5284 
E.W. 


6.1952 


6.6013 


6.2236 

N.S. 1 





E. W. 


N. S. 


E. W. 


N. S. 


E.W. 



The Surveyor's Guide. 



119 



tables of latitude and departure. I 




1 


N". S. 


E.W.I 


N. S. 


E.W. 


N.S. 


E.W. 

45J 


N. S. 


E.W. 1 




44° 


46 


Ui 


45i 


0.7132 


44i 


45i 
0.7040 




0.7193 


0.6946 


0.7163 


0.6978 


0.7009 


0.7102 




2 


1.4387 


1.3893 


1.4326 


1.3956 


1.4266 


1.4018 


1.4204 


1.4080 




■6 


2.1580 


2.0840 


2.1489 


2.0934 


2.1397 


2.1027 


2.1306 


2.1120 




4 


2.8774 


2.7786 


2.8652 


2.7912 


2.8630 


2.8036 


2.8407 


2.8161 




5 


3.5967 


3.4733 


3.5815 


3.4889 


3.5662 


3.5045 


3.5509 


3.5201 




6 


4.3160 


4.1679 


4.2978 


4.1867 


4.2795 


4.2054 


4.2611 


4.2241 




1 


5.0354 


4.8626 


5.0141 


4.8845 


4.9927 


4.9063 


4.9713 


4.9281 




8 


5.7547 


5.5573 


5.7304 


5.5823 


5.7060 


5.6072 


5.6816 


5.6321 




9 
1 


6.4741 


6.2519 


6.4467 


6.2801 


6.4192 


6.3081 


6.3917 


6.3361 




46 


45 












0.7071 


0.7071 




2 


1.4142 


1.4142 
















3 


2.1213 


2.1213 
















4 


2.8284 


2.8284 
















5 


3.5355 


3.5355 
















6 


4.2426 


4.2426 
















7 


4.9497 


4.9497 
















8 


5.6569 


5.6569 
















9 


6.3640 


6.3640 


b7W. 














E.W. 


N. S. 


N. S. 


E.W. 


N. S. 


E.W. 


N. S. 





TAELES OP SUEYEYS 



THE USE OP THE FOREGOING TABLES IN RELATION 
TO SURVEYS. 

They show, by inspection, the alteration of lati- 
tude and departure to every degree on the com- 
pass, and that for any distance not exceeding 
100.000 links. 

In the uppermost rank of every division are 
placed the several angles and their complements, 
to 45°, including the quarter, half, and three- 
quarters of degrees ; and in the left-hand column 
are lengths of the measured lines of the field- 
work, and in the common areas are the difference 
of latitude and departure. 

Examples. 
Suppose the angle to be N. E. 27J degrees, and 
the line in the field measured to 6 chains, and 
it be required to find the Northings and Eastings 
of that station, under 27J degrees, and answering 
to 6 in left-hand column, the number in the com- 
(120) 



The Surveyor's Guide. 121 

mon area, 5.3221, -which shows the Northings ; 
and under 621, (which is the complement to that 
angle) opposite the same number in the side col- 
umn, I find 2.7705, which shows the Easting of 
that station. If the course be the same, and dis- 
tance 60 chains, remove the decimal point one 
place to the right-hand, and the latitude and de- 
parture will be 53.221 27,705. 

And if the line were 600 chains, the course 
remaining the same, the Northings would be 532 
chains, 21 links, and the Eastings 277 chains, 05 
links. 

If the measured line doth not consist of an ex- 
act number of tens, as suppose its length to be 75 
chains, 03 links, or 75 chains, 34 links, and the 
course 27J° ; then under this angle, and opposite 



c. 

70 are 62.091 

6 " 4.435 

0.30 links 0.266 

0.04 " 0.035 



70 chains 32.822 

5 " 2.308 

0.30 links 0.138 

0.04 " 0.018 



Northing. 66.827 Easting 34.786 

for 75 chains, 34 links. for 75 chains, 34 links. 

And so for any other. 

N. B. These tables will answer to ^° or 7J', an 
arithmetical mean between |° and J°, or between 
J° and f °. 



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11 

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13 
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14 
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15 
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16 
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